Relativity sets a limit on the maximum speed matter can travel assuming the matter starts out at a speed less than the speed of light. But, it does allow particles that start out at speeds above the speed of light to travel faster than the speed of light. These particles have been postulated - called tachyons- and there are numerous research papers on them. They plays roles in various models for a unified physics. Unfortunately, in order for the theory to work, tachyons have to have an imaginary mass (the energy of the tachyon has to be a real number forcing the mass to be a pure imaginary number). So this has various implications as to interactions and observability. What is accepted is that tachyons should generate Chernkov radiation which would be light speed electromagnetic waves. Thus, we should be able to detect tachyons by looking for their Chernkov radiation. No such particle have been detected to date in the various experiments that have looked for it.
But given our current understanding of the universe, the only way to take mass from sub-luminal speeds to super-luminal speeds would be through a worm hole - a path "around" space-time that doesn't limit the speed of travel. The first formulation of a worm hole was the Einstein Rosen bridge. Worm holes have not been observed and if they do indeed exist, it isn't clear if matter could survive in a passage through one as they are generally associated with black holes.