The recent report by CERN that they measured neutrinos going faster than the speed of light by about 20 parts per million has made quite a stir. Given that the speed of light is 299,792,458 m/s these neutrinos were going 6 km/s faster than light which could easily be an experimental error.
It would not be a problem for Special Relativity since not traveling at or faster than the light only applies to particles with a real non zero rest mass. A particle like a photon that has a zero rest mass travels at the speed of light. Particles that travel faster than light have been speculated for years under the name tachyons. Tachyons would have a negative imaginary rest mass giving it a real non zero mass at speeds above the speed of light.
Also quantum effects like quantum tunneling provided another way around the light barrier by allowing particles to instantly skip over an area of space without actually going through it while traveling sub light through the space it does go through thus not violating Special Relativity.