Not in the conventional sense, by accelerating with a rocket. Einstein's relativity theory says that you can't accelerate to lightspeed, and the theory has passed every test we've thrown at it.
However, a man called Alcubierre developed a possible loophole. He found a solution to Einstein's equations that described a warp bubble (no, Scotty, I am not joking). You don't move, but you arrange for the patch of space-time around you to move faster than light. There are two problems that I am aware of. One, you need negative mass matter and we've never seen that, nor do we have any idea if it's possible to create. Two, the maths describes a warp bubble already in existence. We have no maths to describe a warp bubble forming or dissipating, and there is some doubt as to whether or not such maths is even possible.
All warp-related scholarship from Wikipedia and the maths is way above my head, so health warnings abound.