In some cases, like a really late Empire Builder, they'll annul the train and turn it short at Spokane and bus the passengers from Spokane, but Amtrak has a pretty fair base of operations in Spokane. This is usually done for Amtrak operational purposes, not so much for passenger convenience. They don't want cascading delays by having a severely late turn of the next trains out of Seattle/Portland. That means the departing passengers are bused, too. Also, if the train is that late at Spokane, the lateness is baked in. Busing is not that much faster than the train Spokane-Seattle or Portland.
If there is some kind of problem mid route so that the line is blocked and the train cannot continue, such as major freight derailment, they'll often stop the enroute trains at stations on either side, bustitute passengers between them and run them back the other way.
Mostly late trains are just late. I've arrived in Chicago 8-14 hours late more times than I've cared to recall. I've eaten the stew many times.