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So, this goes to something so fundamental, I can barely express it. I can only consider it a fundamental breakdown of seemingly intelligent minds.

The Schrodinger's Cat thought experiment ultimately asserts that, until the box is opened, the cat is both dead AND alive. Now, this is obviously ludicrous. The cat either died or lived at some point; someone opening the box and observing it had zero influence on it.

Saying the cat was both alive and dead till the box was opened seems to be some kind of hardware defect in some people's thinking. I mean, with all respect, I don't know how I can be polite about it.

We humans aren't THAT important. Things happen whether we see them or not. I mean, do I really even need to state that?

The question, then: Is Schrodinger's Cat meant to be taken at all physically?

Schrodinger's cat is not both dead and alive any more than an electron simultaneously exists at every point in space. You are using a pop-sci explanation of Schrodinger's cat that indeed falls apart when you dig deeper.

I will use the Copenhagen interpretation of QM for my answer, since it is the most widely used interpretation to teach introductory QM. There are other interpretations that get to deeper meanings, more practical understand of measurements, etc. For that I'll refer you to the other answers, but I am not claiming this is the only way to view this scenario or QM in general.

Schrodinger's cat (or if you hate this example, think "quantum system") is always in a single state. Typically the example says that there is an equal probability of us "measuring" the cat to be either alive or dead once we open the box. Therefore, the cat is in a state that is a superposition of our "life states" $|textaliverangle$ and $|textdeadrangle$:
$$|textcatrangle=frac1sqrt2left(|textaliverangle+|textdeadrangleright)$$

This state tells us that there is a probability of $0.5$ of observing the cat as alive and a probability of $0.5$ of observing the cat as dead. This is because
$$|langletextalive|textcatrangle|^2=0.5$$
$$|langletextdead|textcatrangle|^2=0.5$$

Once we open the box (perform a "life state" measurement of the system), the state of the cat collapses to one of the life states (eigenstates of the "life measurement operator"). So we observe the cat as either alive or dead.

It is important to understand that before we open the box the cat is not both alive and dead. The system cannot be in multiple states at once. It is in a single state, and this state is described as a superposition of life states. Once we open the box the cat is in a new single state which is one of the two life states. We cannot determine which state the cat ends up in though, only the probabilities it will end up in a certain state.

Of course Schrodinger's cat is crazy to think about because we are trying to apply QM formalism to the macroscopic world, but this is precisely how quantum systems work. We can express the state $|psirangle$ of a quantum system as a superposition of eigenstates $|a_irangle$ of a Hermitian operator $A$:
$$|psirangle=sum_ic_i|a_irangle$$
We do not say that the system is in every state $|a_irangle$ at once. It is in a single state (the superposition) that tells us the probability $|c_i|^2$ of the system being in one of the states $|a_irangle$ after making a measurement of the physical quantity associated with operator $A$.

Basically the answer is yes, the cat is both dead and alive. People used to discuss this sort of thing in terms of the Copenhagen interpretation (CI) and the Many-Worlds interpretation (MWI), but those discussions tend not to be satisfying, because both CI and MWI are designed so that in almost all real-world measurements, they give the same predictions. A better way to talk about this is in terms of decoherence.

Quantum mechanics says that the cat is in a superposition of states, alive and dead. Quantum mechanics doesn't impose any maximum size on objects that can be in a superposition of states. Double-slit interference has been observed with large molecules https://arxiv.org/abs/1310.8343 , and there are serious proposals to do it with a virus: http://arxiv.org/abs/0909.1469

However, due to interaction with its environment (e.g., vibrations from the walls of the box, and infrared radiation), the definite phase relationship between the live and dead parts of the wavefunction is lost very rapidly -- the time scale for a cat in a box would be many orders of magnitude too short to allow us to do anything during that time. Once the phase information is effectively lost, it becomes impossible to observe wave interference effects between the live and dead cat.

We humans aren't THAT important. Things happen whether we see them or not.

Right, this was always one of the unsatisfactory things about CI. Decoherence actually happens regardless of whether we observe the object at all. Our interaction with the system would cause decoherence, but so do other interactions, and they do it on much shorter time scales.

I can only consider it a fundamental breakdown of seemingly intelligent minds.

Lots of things in physics are counterintuitive.

Remember Heisenberg's idea that you can't always measure position and velocity at the same time?

So here's an electron, and there's stuff you are guaranteed not to know about it. You can know something about some combination of position and velocity, but that's like having one equation in two unknowns. You know something but you can't solve it like you could with two equations in two unknowns.

Then maybe the electron interacts in some special way. You know its position, and you measure it's velocity. Now you know what it's position and velocity USED TO be, but no longer. For a moment there, you knew.

Before you measured, you didn't know. You had a probability distribution which gave information you did know about it, but you couldn't know it all. Then you knew. And a moment later you didn't know again but had a new probability distribution.

And Heisenberg says there's no way you could know more.

Here's the point -- we naturally want to think that there is a single reality going on that we can't know about. And there could be. But science is about what we can measure. If there's no possible way to find out about that hidden reality, why should we care about it? If all we know about is probability distributions, why not proceed as if the probability distributions are all that's real?

Logically that works just fine. But people don't like it. But logically it works just fine.

If it's things we can't know about, why choose which way to think about it? If somebody wants to think that invisible undetectable elves are making electrons move the way they do, according to probability functions, why argue with them? Their explanation fits the facts as well as yours does. You could argue that yours is simpler. But so what? Their explanation makes them feel better, and your explanation makes you feel better.

Arguing about explanations for QM which go beyond QM is not physics. It's philosophy. Metaphysics or something. Unless we find a way to find out the things that Heisenberg says we can't find out, it doesn't matter.

But -- Heisenberg doesn't really say you can't know those things. Just that you can't find them out using the things we know about in physics so far. Maybe someday physics will advance to the point that those things do become measurable.

I think they can't be measured using leptons, hadrons bosons, and the four fundamental forces. (Is it still four forces, or just three, or two? No matter.)

Maybe someday physics will discover new particles and new forces that make it possible. But for now, physics isn't about explanations for QM that can't be measured. That all give the same results.

Forgive the length. I find Schrödinger's cat is much easier to make sense of as a journey through QM, rather than just a few equations that someone says "solves your problems."

Schrödinger's cat was definitely meant to be taken seriously, in that it was intended to be a serious challenge to naively applying the Copenhagen interpretation to macroscopic objects.

The general challenge brought forth is that constructions like Schrödinger's cat have so many particles, thus an enormous state space, such that simplifying it down into binary states like "alive" and "dead" yields incorrect results.

The real trick to the experiment is the element which is oft overlooked. It's not the cat, or the radioactive isotope. It's not even the box. It's the detector inside the box. You question whether we are "special" enough to collapse a waveform. It's actually not us opening the box that will cause a collapse, but the detector. It's job is to take a quantum level event of "a particle that has a 50% chance of decaying during the experiment" into "a classical measurement of whether the particle decayed," which we then use to signal the machine to use the hammer to smash the vial of poison Just putting such a detector in a box doesn't make it any less of a detector. It's still doing the classical thing.

So what if we wanted to treat the detector as a quantum thing? After all the point of Schrödinger's cat is to poke and prod at what happens if we try this?

Well now we have to be a bit more careful. We have to consider not only the state of the cat and the isotope but also the state of the detector. And the detector seems to be the tricky bit, as it's job is to go quantum to classical, and that makes it interesting.

So what's such a big deal about a quantum thing anyways? Why do we need to have such a confusing model of the world. For the most part (read: everything you or I will experience in our lives unless we become a physicist or some flavors of engineer) is well described with "classical" behaviors. These don't confuse us. However, there are some situations which arise at atomic scales which simply act "odd." We find situations where particles appear to teleport through walls or take two paths at the same time. To make sense of those, we needed new math.

The new rules are, statistically speaking, a superset of the old ones. In most situations, we have lots and lots of particles. We don't know their state, but we can know probabilistic what their state distributions look like. If you run these new rules over large sets of particles for long periods of time, you get the same results you expected from classical thinking (okay, maybe "long by quantum standards" milliseconds are a long time for many quantum systems!)

More to the point of Schrödinger's cat, these new rules obey a principle known as "superposition." In Aaron Steven's answer, he was very careful to point out that the cat exists in exactly one state at all times. There's a good reason he was so careful there. When we write something like $|textcat_initialrangle=|textaliverangle$ or $|textcat_finalrangle=frac1sqrt2left(|textaliverangle+|textdeadrangleright)$, we are describing the one and only state that the cat is in. However, by the rules of superposition, we can figure out the state the cat will be in by looking at each branch of an addition, one at a time, and then add them up later. This is convenient for you and I, because we are much more comfortable thinking through what happens to an "alive" cat or a "dead" cat, rather than trying to handle some complex mathematical equations that links both. The fact that QM wavefunctions have this superposition property lets us do this rigorously.*

And, indeed, for observations, we arrive at the same thing Aaron described. The probability of us observing the cat as alive is 50%. It behaves precisely as if the alive/dead variable was merely unknown until we open the box. There are no surprises there.

But the story isn't done, because there's other things we can do to the box.

There are operations we can do which don't operate in such simple ways as our classical observations do. Quantum operators are fascinating linear functions which can do things we don't always expect. After all, that's why we have QM. And this is why the sensor matters.

We can operate on the cat/box/sensor/particle system with a quantum operator if we like. And, if I may be a bit informal with it, the system after interaction might be $|textcat_afterrangle=a|textaliverangle+b|textdeadrangle+c|textwierdrangle$, where $a$ $b$ and $c$ are just real numbers. The $|textaliverangle$ handles the cases which are handled intuitively as having an alive cat, $|textdeadrangle$ handles the cases which are handled intuitively as having a dead cat, and $|textwierdrangle$ lumps together all of the really wonky cases where quantum mechanics says one thing where our intuition says another. One of the great things about the bra-ket notation that physicists like to use is I can use it to correctly capture a system, even when using really oddball states like "wierd."

So now we come back to the detector. This detector could have been any system really. There's more interesting things to throw into a box with a cat, but the experiment calls for a detector. And, hand-waving emphatically, one aspect of a good detector in physics land is that it minimizes the probability of any weird things happening. Using the above equation, we try to design sensors in such a way that, for any interaction one may wish to do with the system (opening the box, or any quantum operator), the constant $c$ in $c|textwierdrangle$ is vanishingly small ($capprox 0$). A sensor which doesn't have this property is a pretty poor sensor, and I would no longer be comfortable with the intuitive idea that it "detects" the radioactive isotope decaying.

So this detector (which itself has a macroscopic state) was designed to make it incredibly hard to operate on the system in any way which distinguishes it from the simple alive or dead cases which were well described by being "unknown" earlier. Its job is to make the whole "collapse when you open the box" idea defunct, because the observation already happened inside the box by the detector.

Now you can construct more interesting experiments with things other than nice clean detectors. And you can start to see quantum effects at the macroscopic level. There's an entire approach to QM around studying "decoherence" which handles this in a statistically rigorous way and does a good job predicting the results of more odd systems that permit more $|textwierdrangle$ through by design. For example, there's a whole approach of using "weak measurements" which are measurements designed to not disturb "weirdness" that was already happening in the experiment. But in this case we can comfortably say the detector "collapsed" the wave form. And, approaching the topic through the idea of decoherence, we can even show why that term is valid: we intentionally designed the detector to "collapse" the weird part of the waveform into a vanishingly small part.

So never forget the detector. It was a small part of the experiment, but it turns out to be where the devil decided to put all his details.

^{*. As a perhaps useful aside, the decomposition itself isn't all that important. This could have been $|textcatrangle=aleft(|textmalerangle+|arangle femaleright)$, describing what happened to the cat if it was male or the cat if it was female. The math would actually end up right either way. However, by selecting states which are convenient to the human doing the math (alive and dead), it becomes easier to leverage the superposition principle to actually start picking away at the problem, rather than merely developing new bases.}