Special relativity is essentially an explanation of why the speed of light is a constant regardless of how you measure it. That is, if you're in a train moving at half the speed of light relative to the ground, and someone fires a laser in the same direction as the train from the last station you passed through, that laser beam will move towards you at at the speed of light. If you fire a laser back, it will reach the station at the same time as the laser from the station reaches you (as seen by an observer in the station).

This makes no sense unless the speed of light is a fundamental physical constant, so that motion in general depends on the speed of light, which is what special relativity postulates.

Now there are ways to have a special kind of non-locality that do not violate special relativity - you can have phenomena that happen at infinite speed, but only if they do not carry mass or energy or any information at all. The common interpretation of wave-function collapse is an example of such a phenomenon.

I'd also note that the famous E=mc^2 is also a limit on speed, since kinetic energy (mv^2/2) is part of the total energy of an object.