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I have created a simple data acquisition module for measuring 5V signals. There are 18 analog inputs on this board, all of which are a copy of the schematic below.

Each input has an op-amp (MCP604) with unity gain. This op-amp should work to within 60mV of VCC, but in half of my circuits, it won't get any higher than 4.3V. (Notice the 100K pull-up to 5V, the input pin disconnected, should output very close to 5V.) In the other half, the circuits operate just fine. I have two of these boards built, and most (not all) of those circuits are limited to 4.3V. Under 4.3V, the circuit will behave in a linear fashion.

It doesn't appear to be a glitch at power-up, the circuits that work always work, the circuits that don't always don't.

I've cut the trace at the output of the amplifier to see if the A/D converter was clamping it somehow. No effect.

I've tried adding load to the output, with different values of R50, 10K to infinity. No noticeable effect, maybe a couple of millivolts.

Probably an unnecessary detail, but for the sake of completeness...
I need to get the zero input of the signal within the linear range of the amplifier, so I boost it slightly with the 20K/100K voltage divider. This lifts a 0V signal to 0.833V. The device connected to this circuit may provide an input voltage of slightly less than 0V (sometimes a low as -0.5V).

You are exceeding the amplifiers input common-mode range.

It has rail-to-rail output but not rail-to-rail input. The allowable input range is -0.3V to +3.8V when fed from a 5V supply. When configured as a unity gain amplifier the output range will be limited by what's acceptable at the input.

If you exceed the CM input range the output can do almost anything but modern opamps are usually well-behaved. It looks like this one just clamps the output voltage. However, this doesn't explain why some of your boards work.

You can't use it as a unity gain amplifier in this application.

If you made the gain about 1.5 or more it would work as the input would never get above the 3.8V limit before the output hit the power rail. Or find a pin-compatible amplifier with a wider range of input.

Since you have R53 and R54 already placed on your board you could put appropriate values in there to provide the 1.5 gain (e.g R53=10k, R54=4.7k).

Also since you have R51 and R52 you could add resistors there to attenuate the input by the same factor so the overall system works as designed.

Datasheet

You are misusing the device, as a unity gain buffer there are restrictions:

Read the datasheet for the device and particularly note 4.1.3

This tells you that the device cannot be used rail-rail Vin.

^{simulate this circuit – Schematic created using CircuitLab}

Output Impedance reduces with gain using negative feedback.

However, when driven towards the rail there is zero voltage gain to go any higher. It will just rely on the RdsOn to pull to either supply rail. Thus you now recognize a resistive load will reduce the output.

If you read the datasheet closely on Rail-Output...you should recognize that you are experiencing a "load regulation error" from this impedance divider relationship.

For instance, the output voltage swings to within 15 mV of the negative rail with a 25 kΩ load to VDD/2.

Therefore, you will want to choose >1M load for any Rail-to-Rail Output.

Quiz: If your "no-load" output = Vdd= 5.00V and 50 Ohm to 0V with an output = 4.28V what is your open loop source impedance?

The input is not R2R so a unity gain differential amp design is needed so the Vcm becomes Vdd/2.

The required output current is too high. The datasheet says 22mA short-circuit current. 5V across 99 ohms (R49 + R50) demands 50mA and that's ignoring the fact R53 also needs to be driven.

It's also not a good idea to drive an ADC through a divider. Lower resistances help, but then you run into theses drive current issues. Why are your resistor values so low to begin with?