If an atom was to travel at the speed of light (although impossible), wouldn't the nucleus break apart? I say this because the particles holding the protons together in the nucleus wouldn't be able to reach them and keep the atom together. So even if special relativity was wrong, wouldn't it still be impossible for an object with mass to travel at the speed of light.
Of course, this is impossible, but you can use it as a thought experiment. You're looking at the behavior at a singularity, so the way to deal with this is to investigate the limit. In this case, you can do that intuitively without the mathematical formulation: Because of the relativity principle, no matter how close you are to the speed of light, for you light still moves with the speed c in either direction. So there is not even a hint of any change. Therefore, in the limit, the to and fro of particles still works the same.