Your assumption is that the matter which accelerates to the speed of light remains on this plane of reality. Of course that would be impossible, as you say. So the speed of light cannot actually be acheived, but by changing planes, then traveling a distance, and changing planes back, one can create the effect of having gone the speed of light. This concept is like hyperspace in Star Wars, or Warp speed in Star Trek. Both methods behave in a manner that seems to be many times the speed of light.
JediMusician> What "planes"... ?
We have detected no more dimensions in the Universe tha four. If there existed more, then matter would enter and exit these planes (because they CAN) and we'd see it.
Warp speed ala Star Trek is not about "other planes". It deals with manipulating gravity, so you compress spacetime in front of a space-craft and stretch it out behind it, effectively creating a space-time-bubble. Relatively to the space-time bubble the ship would not exceed the speed of light, but the space-time bubble itself would. We know the Universe is expanding at speeds greater than the speed of light, so spacetime itself can do that.
How to make warp-bubbles? Wellll... you need matter with some rather peculiar properties (negative densities), but hey... Let's cross our fingers that the new LHC in CERN will finally kill String-Theory so all the bright physicist start doing some useful quantum gravity and then... who knows? 😄