Our current understanding of physics puts a hard limit on matter traveling at or above the speed of light as your mass increases as you approach the speed of light requiring an ever greater (infinite amount of energy) to get you there. With that being said, we still don't have a perfect understanding of physics, so there is wiggle room. However, our models are really good both on the drawing board and in observation, so it looks unlikely at this stage (I'm not a physicist, but read the dumbed down books they publish for the masses).

That's not exceeding the speed of light. That's changing the distance you need to go so that you can get to where you want to be at sublight speeds in reasonable time.