Yes, it's possible. Quantum particles travel faster than light because when they "tunnel" through a barrier, it takes place faster than light. And it's been measured. Plus, quantum particles are able to communicate with each other instantaneously over distances.
According to the special theory of relativity, you can't start out with a speed less than that of light and go faster and faster until you pass the barrier because time stands still at the speed of light itself, while length contracts to nothing. But it's only the actual speed of light itself that marks this forbidden territory. If something existed that always traveled faster than light relative to us, it would travel backward in time.
There are, in fact, solutions to the equations of the special theory corresponding to particles which always travel faster than light. They have been called tachyons. Though still hypothetical and embarrassing to most physicists, they are not ruled out by the special theory.