My personal experience with the DC - NY route (which also has at least one bus an hour) is that nobody looks at the time of the ticket. I've tried it several times (taken an earlier bus, or a later bus, than my ticket) with no problems.

For what it's worth -- if you're anywhere along the NE corridor, BoltBus (part of Greyhound) and MegaBus are both cleaner and generally terminate in less-odd parts of town. MegaBus does stop at intermediate cities, while BoltBus operates out of NY with direct trips to Philly, Boston, and DC -- but I find it's hard to beat their fares. BoltBus also has a similar policy with respect to the time listed on the ticket, space permitting.