I never thought I'd live to see the day someone confused DC for being in Washington state. Though, OP does have a point. If you have either Enclave/BoS connections or are really good at stealing things, you could theoretically make the trip from DC to Vegas in two days flat. I speak, of course, of hijacking a Vertibird. True most wasters have no hope in hell, but if you have the right connections or a screwdriver and some skill at vehicle theft you oughtta be able to do it.
I very much doubt a wastelander is going to figure out how to fly a Vertibird before crashing and dying.
Making the trip on foot is not impossible, but it would be very difficult, especially once you reached the former heartland and the epic dust bowl conditions likely to be present there. Now imagine http://www.paranormalknowledge.com/articles/wp-content/uploads/2008/12/dustbowl.jpg, only all that dust is radioactive, and mutated animals or raiders are stalking you somewhere in it.
The actual distance isn't that bad, and you would have no need to move at a break-neck speed to reach New Vegas from the Capital Wasteland. 4 years have passed since FO3, and the 2500 mile journey could be completed on foot if the wastelander averaged a little over 2 miles a day. That's a pretty leisurely pace, wasteland conditions or not. If they travel 10 miles once every 5 days, which wouldn't be that hard either, they could even rest/scavenge/resupply for nearly a week in between trips and still make the trip in less than 4 years.
I'm even writing a comic to explain my New Vegas character's backstory that involves this (as she comes from Capital Wasteland originally). If you imagine a character left the DC area 6 months to a year after the end of Fallout 3, they have plenty of time to reach Vegas and become the Courier before the start of New Vegas's storyline. (My character for New Vegas will be a new one, and not related to the Vault Dweller in anyway.) Travel an average of 5 miles a day and you still have time for extended layovers of months or weeks in certain spots.
The two big obstacles are going to be the Appalachian Moutain range, which while not too difficult, will definitely slow a traveller way down, and the radioactive dust bowl in the midwest. The Mississippi River, averaging only 9 feet deep in most places above New Orleans, has likely dried up, or lowered to the point it could be waded across with no need of a boat or raft. Keep your supplies up for the trip, watch out for raiders, and you're golden.