Nanette,
You should be fine. While excessive delays do sometimes happen, the connection between the Empire Builder (the train you're taking from Seattle to Chicago) and the Capitol Limited (the train from Chicago to Washington, DC) is relatively reliable. If the Empire Builder does somehow get delayed such that you miss the Capitol Limited, Amtrak will most likely try to find room for you on the Lake Shore Limited from Chicago to New York, and get you to Washington DC via a regional train that evening. You'd be delayed arriving into DC, but wouldn't have to spend the night in Chicago.
Regardless, Amtrak takes responsibility for you if you do get delayed in Chicago, and will take care of getting you a hotel.
Again, this is the exception to the rule, and for the most part, the Empire Builder connects with the Capitol Limited without incident every day.
Rafi