The situation is not really very analogous to speed of sound. We knew that it was possible to go faster than speed of sound since bullets, etc. were doing it. The question was whether man and machine would be able to survive the dynamic pressures in the transition... several vehicles did not survive the attempt.
Unlike that, the state of physics right now would indicate that it is not at all possible to exceed light speed, certainly by brute force. We don’t have any convincing examples at this time. However, given that we have clear understanding that the mechanism for that is that space-time is “curved”, that leads to the intriguing possibilities of attempting to “jump over” areas of space, instead of traversing it... “worm holes” and such thoughts of intentionally curving space somehow. I don’t see any likelihood that technical means of accomplishing that can possibly come about until we have massive, controllable energy sources, likely fusion power plants. And surely, it is possible that somehow we can discover some way to negate the increase of mass as velocity increases somehow, but again, there is no proposed mechanism whereby that could occur.
Two other things: it appears that a “negative mass” object may be able to satisfy the relativity equations at “faster than light” speeds, but that can’t help us to, say, launch a space ship to Alpha Centauri. There also exists the intriguing possibility of “entangling” two particles at very great separations, and thereby “transfer information”, from one place to another, at “superluminal” speeds, That really doesn’t satisfy me as being “faster than light” travel, if only because my soul would not be able to partake in the trip.
The series was set in the 24th century, but the tech toys that were just imaginary in 1966 have become real or nearly real. These series fired the imaginations of many people and who knows what may come of it?
The current thinking is we can't go faster than light and that is certainly true with our current tech. But, in 50 years, who knows? We may have transporters and replicators and holodecks and (fill-in blank).