My experience on the long-distance trains (Capitol, Lake Shore, California Zephyr, City of N.O.) is that dinner is normally served 5:30 PM until last call around 8:00-9:00 PM. I've found that breakfast normally starts around 6:30 AM until around 9:00 AM.

So, as long as the train is on-time into WAS from NYC, you should have the opportunity to grab a late dinner - just check with your sleeper attendant as soon as you board. Once the conductor has lifted your ticket, you may be able to proceed towards the diner if there is openings at that seating time. Otherwise, you'll need to wait until last call is announced over the P.A. system. Your car attendant can check with the dining car staff.

At breakfast, you can head to the diner as soon as meal service is announced, sometimes earlier if a set time is in place - ask the dining car steward when they plan on opening up in the morning. One good thing about trains running tardy is you may get an extra meal. I used to de-train on the westbound (29) Capitol in Toledo, Ohio - scheduled for 5:30 AM. The few times the train was running late I was able to enjoy my complementary breakfast before arriving - nice!

Just don't forget to tip both your car attendant and dining car server - it goes a long way towards them taking care of you.