I think the real question here is "why is Newton's second law invalid at high velocities?" Indeed, it seems arbitrary to declare that a rule like F = ma is valid at low velocities, but not when v becomes large relative to c. In fact, this seems to violate the principle of relativity itself, giving mathematically "preferential treatment" to objects traveling at certain velocities. However, this is not the case. In my opinion, the best way to understand why is to think carefully about which of the foundational concepts one learns about in mechanics (i.e., force, mass, energy, momentum, etc.) are the most "fundamental".
For example, in the Newtonian picture of mechanics, "force" is the fundamental quantity that determines the motion of all bodies relative to one another. However, even from a classical point of view, this is not necessarily the most convenient mathematical formulation. As is well-known to anyone who's taken kinematics, conservation laws frequently provide a vastly more efficient approach to solving problems than a direct application of Newton's second law.
Now, in the classical world, these conservation laws can be shown to follow from F = ma, and are thus "subsidiary" in some sense--in other words, you will never get the wrong answer by just applying F = ma (assuming you can successfully carry out the math). However, the utility of concepts like "energy" and "momentum" led physicists to wonder if perhaps these quantities are actually more "fundamental" than force. Classically, this line of thinking led to re-formulations of mechanics in terms of Lagrangians, Hamiltonians, and fields, each of which have their own "domains of utility" in which they're useful descriptions of nature.
All of the alternative schemes mentioned so far can be shown to be equivalent to Newton's original formulation of mechanics. However, when combined with a careful examination of Maxwell's equations for the electromagnetic field, taking a "momentum-first" approach to physics leads to a point of departure from Newton's second law (the three-dimensional version, that is). Maxwell's equations respect certain symmetries (called Lorentz transformations), and requiring the equations to apply unmodified in every inertial frame of reference implies that momentum transformations between frames must also respect these symmetries.
In other words, it's a question of where we "begin" mathematically--do we postulate that F = ma is always true and work from there, or do we take conservation of momentum, the electromagnetic field, and the principle of relativity to be fundamental and derive consequences from these? If we choose the former, then there's nothing preventing me from accelerating to the speed of light. If we take the latter approach, the speed of light naturally emerges as a strict upper bound on the relative speeds of any two observers.
Ultimately, the question can only be settled by experiment. At this point, the verdict is pretty much in.
Relativity wins.
