An American traveler with a valid rail ticket can still be turned away from a European train. Whether a reservation is compulsory depends on how that country's operator manages capacity.
Two different models of selling a train
One model treats rail like a bus. A ticket is valid for a route and a period, seats are unallocated, and the operator accepts whoever arrives.
The other treats it like an airline. A specific train has a fixed number of seats, each sold once, with the fare varying by how full the train is.
Most networks use both, applying the open model to regional services and the reserved model to long distance and high speed trains.
Why high speed services reserve
A high speed train has a fixed formation that cannot be lengthened, and standing passengers at those speeds are not acceptable for safety and comfort reasons.
Selling exactly the number of seats available is the only way to guarantee that, so boarding is controlled at the door on many services.
It also enables yield management, letting the operator sell early seats cheaply and later ones dearly, which is where most cheap advance fares come from.
Passes do not remove the requirement
A rail pass covers the fare component but not the reservation, which is bought separately and priced per train.
On heavily reserved networks those reservation fees accumulate quickly, which is the most common reason a pass underperforms against point to point tickets.
Pass holders also compete for a limited pass quota on some trains, which can be exhausted even when ordinary seats remain available.
Cross border trains follow the stricter rule
A service crossing between a reservation-optional country and a reservation-compulsory one generally applies the stricter regime for the whole journey.
This catches travelers who have used the same network domestically without reserving and assume the international service behaves identically.
Night trains are compulsory almost everywhere, since a berth is a physical allocation rather than a preference.
How to check before the platform
The operator's own booking engine states whether a reservation is required, and that statement is authoritative in a way third party sites often are not.
Where reservations are optional but the train is busy, buying one anyway is inexpensive and avoids standing for several hours.
The practical rule for planning is to reserve every long distance leg in advance and treat regional connections as walk-up, which matches how the networks actually work.