Because everything is relative so is mass. Time becomes relative, and if time is relative as well as distance. And damian was on the ball. Because you need more energy content in order to travel, you mass must give way to provide that. (Such as light. Light is "massless" because all of it's mass is being converted into motion).

And there are ways to test this. In particle colliders, you can approxiamate the mass by an object falling. As objects approach the speed of light, they fall at a more acute angle, meaning that their mass is more. Also, you can hit particles with light and measure their movement. A heavier particle will move a little.

But again, you'd have to be traveling at 99% the speed of light to tell the difference. I believe that an electron traveling at 99.1% and another electron moving at 99.2% will have a 120% difference in weight. (I can't remember the figures)