I will say this. Our current knowledge of physics is not enough to rule out FTL travel. But our current knowledge is enough to say that the problem is VERY difficult. I seriously doubt that 2 or 3 hundred years in the future is going to be enough. Add in the fact that we haven't (to our knowledge) been visited by any cultures that have solved the problem, and what you see is the problem is pretty frigging hard. So hard in fact that I don't think that it is reasonable to expect anything like a warp drive solution in less than a 1000 year time frame. I hope I am wrong.
IMO, the biggest blunder Roddenberry made in trying to make Trek plausible is not placing it far enough into the future.
I will be merciful, and quick.