This may not happen for a very long time, Star Trek's timetable may be off, but assuming the world lasts long enough, this too will be discovered by scientists.
I think pretty much anything can be discovered. The rate of research increases dramatically as time goes on. A thousand years ago gunpowder didn't exist. Two hundred years ago man could never fly. Fifty years ago man was just learning to leave the atmosphere. We haven't reached Mars yet, but with time we likely will.
Frankly, I'm not so sure its WORTH trying to develop FTL, since I don't believe there is anything else out there in terms of alien life, and I'm not sure we'll ever reach a planet we can live on (Given enough time, of course, technology should fix this problem as well) but I'm sure it'll happen... eventually...
I bet by 2200 we'll have it but who knows?