Due to special relativity, this is not the case. At this point, many readers will question the validity of any answer which uses advanced, intimidating-sounding physics terms to explain a position. However, it is true. The relevant equation is v/c = tanh (at/c). One will find that in this equation, tanh(at/c) can never exceed or equal 1. This means that velocity can never reach the speed of light, regardless of how long one accelerates for and the rate of the acceleration.

Unfortunately, that's only half of the truth. You're probably thinking of an object that accelerates at a constant rate, then abruptly stops accelerating when it reaches the speed of light. The problem with this idea is that it violates relativity. The real reason that we say that the speed of light can't be exceeded comes from the Lorentz factor, calculated as (1-v^{2}/c^{2})^{-1/2}. This factor is used in special relativity for many different purposes, but most notably for time dilation. Essentially, while a moving object feels the same force regardless of speed due to the principle of relativity, the rest frame will see it accelerate progressively less and less according to the Lorentz factor. Since it approaches infinity as the object approaches the speed of light, that speed is never exceeded to the rest frame.

*mic stays stationary and earth accelerates upwards towards it*