That is true, in a way.
However, if you studied the History of physics theory, you'd see that usually, the laws stated by the great minds (Newton to here, mostly) were a subset of the laws found by their succesors (IE: Einstein, Schrodinger, etc.).
So, even when it is certain that there are many things we don't know, it is most likely that that wich we do know is still contained in the laws that we don't know.
This means that it is most likely impossible to reach the speed of light through normal acceleration. Wheather or not we may find another way to move that is different from F = m * a remains to be seen, but I get the feeling that if we do, the OP's question will become irrelevant.