Possible problems
1) It's tired, and needs its beauty sleep
2) You got the "Cinderella" revision - it turns into a pumpkin at the stroke of midnight
3) You have inadvertently turned on the Airport Midnight Express setting. It breaks out of jail once every night.
What happens if you restart the router once it disconnects, does it then dial back up and work?
What does your ISP say? Do they have a limit on dial up connect time? Do they force a reconnect at a specific time?