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A German Sparpreis-ticket is (for long-distance trains) only valid on the trains indicated on the ticket, but a seat reservation is optional. In case of overcrowding, a Sparpreis ticket holder can be denied boarding to the indicated train. What rights to alternate transportation or restitution does the traveller have in this case?

Based on common sense and consumer protection, I would expect that if Deutsche Bahn cannot deliver what the traveller paid for, then either the ticket should be valid on a later train or should be reimbursed. Therefore, I find the accepted answer on community.bahn.de (which states the opposite) hard to believe.

What are my rights if I am (to no fault of my own) denied boarding to the train indicated on my ticket?

The chances of you being "denied boarding" are very low. There is no pre boarding check, nobody is standing at the door counting passengers. So there is nobody to deny you boarding.

What is possible is that the train is physically so full that the train cannot leave, and when it does happen it usually escalates to the point where a train will be significantly delayed. And as soon as the delay is significant enough that you can expect to be more than 20 minutes the "zugbindung" (train binding) is cancelled.

I have been on a train that was so full it wouldn't leave, and the conductor started asking people that were standing to leave the train. We ended up with a delay of one hour, and I wished DB would introduce compulsory reservation for its long distance trains.

I always reserve seats. However to someone without a reservation I would suggest you make the removal of the "zugbindung" a condition of volunteering to leave. All the conductor needs to do is write it on the back of the ticket.

Regarding the answers on the Bahn Community forum:
https://community.bahn.de/questions/1528379-aussergewohnlich-hohes-reisendenaufkommen-mitfahrt-garantiert

In this case the person who is asking the question has not been refused access. He or she however fears they will have to leave the train and asks if they can assume that the "zugbindung" has in such cases already been lifted a priori, or if they can change the ticket.
The DB answer is that this is not the case, and that they still will have to present themselves to the train they booked.

There are related questions on the forum, and one is here:
https://community.bahn.de/questions/1487525-zug-ausgebucht-trotz-sparpreis-andere-verbindung-wahlen
There the answer (also accepted by DB) is that if you cannot use the train you wanted to take that of course you can take another.

DB has no choice here. If they sell you a ticket they have to transport you. If they can't transport you with the train you ought to take they have to offer you a different one. And you may even be able to claim compensation for the delay...

In general, this is driven by the practicalities of rail travel.

First let's exclude cases where the passenger has missed the train.

When the carrier imposes a restriction that the transport will only work on a certain run (like nearly all airline tickets), and the carrier refuses to provide transport on that run... then the carrier is obliged to provide reasonable alternative transporation.

Airlines are obviously good at this, because they are also formal. They formally reticket you (because load management is critical, no standees in air). And you need a document so you can exit and re-enter security.

With railway, well, they are also pretty good at this, but they tend to do it much more informally.

At minimum, they may simply tell you to use that ticket on the next available train. What happens then? Most likely the conductor has already dealt with 10 other parties with the exact same story, so that proves it's true. Or if you're the first, it's possible you are lying and were late due to your own fault, but the conductor knows company policy is to let you ride the next train with available (about-to-be-wasted) space, and here you are. It's a distinction without a difference, and certainly not worth making a huge scene.

Also, this is not his first rodeo: he knows the 1:00 train often overbooks so your story is plausible to him. He may even be aware of an equipment problem (car removed from the train; car present but with inoperative A/C) that caused the overbook.

* For Amtrak/Via long-haul, this tends to be a huge production because of the need to book hotel if taking next train, or airline tickets if your travel is time-sensitive or if your multi-day train can be overtaken. Fortunately this mainly happens at hubs: NYP, WAS, CHI, LAX, Toronto which have enough staff to handle a mob.