Now if you were traveling along with the object, even at the speed of light, it would appear to have its ordinary, rest mass. It's only to the outside world that it appears to have a greater mass. In other words, it's only if the object is moving *relative to you* that you see a difference--which is why we call it relativity. If you're moving along with the object, we say that you are in the object's reference frame. A stationary observer has a different reference frame.
Of course, it may be possible that special relativity is wrong, and something else happens at extremely high velocities. But big particle accelerators like at Stanford and in Switzerland use special relativity every day, and it's been perfectly correct even for the fastest particles we can accelerate.