At least, that's what Einstein's equations tell us.
According to the mass-energy equivalence theorem, If your energy increases, your mass would increase as well. But what happens when an object is massless?
Consider the right triangle below. If your mass (m) is zero, your energy (E) and your momentum (p) would be equal. This means that your energy is equal to the speed of light, and therefore, you move at light-speed.
HOWEVER, if your mass have a value, even a miniscule one, part of your energy will always be directed towards increasing your mass, not your speed. You'd end up becoming heavier, than faster. This is why when you try to speed up to reach the speed of light, you would just increase your mass, and you'd actually go slower over time.
Source:
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