The answer is no, but there are some subtle concepts here. Relativity says that, in a vacuum, nothing can locally move faster than the speed at which zero-mass particles move. Light is composed of zero-mass particles called photons, hence this limit on the local speed of motion is typically expressed in terms of the speed of light. However, any particle of zero-mass should move at the same speed in a vacuum: if neutrinos are of zero mass, then they move at the same speed as the speed of light.
If there isn't a vacuum, then things can move faster than the speed of light, even on a local basis. For example, Cerenkov radiation, the blue-tinged light emitted by the water in nuclear reactors, is the shock-wave of radiation produced by a charged particle moving through an insulator at a speed greater than the speed of light in that insulator.
On a non-local basis, the time taken to travel between two spatial locations is dependent upon the geometry of the path taken between those two locations. If one creates or chooses the appropriate path, one can complete the journey before photons of light taking a different path. This is demonstrated by Miguel Alcubierre's model for a warp-drive. The basic idea of the warp drive is that it creates a bubble of compressed space which the space-ship travels within. The space-ship reaches its destination very rapidly because it travels a very short distance, not because it locally violates the speed of light.
Things can also change faster than the speed of light. For example, a pair of galaxies can recede from each other faster than the speed of light, under the expansion of the universe. The recession velocity v of a pair of galaxies separated by a distance d is given by
v = H d,
where H is the Hubble constant. Hence, any galaxies separated by a distance greater than c/H (where c is the speed of light), will recede at a rate greater than the speed of light. The recession velocity of the galaxies is due to the expansion of space between the galaxies, not due to the motion of the galaxies through space. There is no limit at all on the rate of change of spatial distance.