In a word, no. It is understandable to think that way however, when you drop an object on Earth, the speed of the object increases as it falls. One would then think that if a photon was approaching a black hole head-on, it too would gain speed, thus violating the speed limit, right?
The problem is that the speed increase we normally see in a falling object is only the result of what actually happens. As something falls into a gravity well, it gains energy. For large, slow moving object that we are used to seeing, this gain in energy manifests as a gain in either kinetic or potential energy. Since something cannot fall to a lower potential and gain potential energy, it must gain kinetic energy. This means its speed will increase. However, for relativistic objects and/or photons, there are two things that must be considered.
First, when we add velocities, it is not simply $v_A+v_B=v_C$, the tested and true equation for adding any two velocities is:
You might notice that this means one cannot add any two velocities less than or equal to c and get a value greater than c.
The second thing is that when a photon (light) falls into a gravity well, its gain in energy is not limited to increasing its speed. Because the energy of a photon is $E=h\nu$, where $\nu$ is the frequency and h is Planck's constant, this means that when its energy increases, the frequency of the photon can easily increase as a result. This is, in fact, what happens. Because a photon cannot go faster, it ends up increasing its frequency to accommodate for the increase in energy. This has been tested and shown to be what happens when light is affected by a gravity well.
That being said, it is natural to question counter-intuitive statements and I'd be surprised if most good physicists didn't think something similar to this at one point in their life. A healthy scientific compass is OK, but for the future, if you come across something where you find yourself thinking "this is so simple, comes right from common sense, and disproves one of the most fundamental tenets of modern science. I can't believe no one has thought of it before", stop. That is a good indicator that you may be on the wrong track. Millions of the world's most brilliant minds have studied, experimented on, and longed for the opportunity to disprove stuff like this. Even the smartest person of all time would be outweighed by statistics at this point.