It's really a very good idea. Almost everyone offers a 6am flight, which means you need to be at the airport at 5am minimum, if you live an hour away, that means leaving your house at 4am, so you probably have to get up around 3 or 3:30. Some people will book a hotel room close to an airport for an early morning departure, but then that is an added expense. If your flight leaves at 4:10am, you have to be at the airport at 3, which means leaving your house at 2, so if you're any kind of a night owl, it's not all that bad. You get in early to your destination, if you're able to check into your hotel early or wherever you are going, you can get sleep till about noon. I'd rather not go to bed, than have to get up at that ungodly hour and still feel tired when I get where I'm going.
Also, I think the San Juan flights at 2:55am offers several morning connections at JFK
. They have offered that flight since they started that route so it must be doing pretty well.