Yes we do really know, and the answer is that causal influences cannot travel faster than the maximum speed. Light in vacuum travels at this maximm speed, so we say that nothing can travel faster than light. This means, to be precise, that a causal influence cannot move past other nearby things faster than light would move past them.
There are plenty of examples of phenomena where some sort of definable spot or intersection or whatever moves faster than light, but these do not transmit information or causal influences.
Gravitational phenomena obey the speed-limit, but sometimes people fail to understand the possibility that there may be more than one route between given places, so if an influence took the shorter route it can arrive faster than a signal such as a light signal that went by the longer route. There's nothing particularly suprising about that, but when these routes are connected with spacetime being distorted into strange shapes, then sometimes this may catch us out.
Stuff falling towards the singularity in a black hole approaches but does not exceed the speed of light. For example, it won't overtake a light signal sent on ahead of it. However, the singularity itself is a place where our physical understanding is almost certainly incomplete; very likely something is going on there which has to take quantum theory into account and we don't know what.
You sometimes hear of work in quantum field theory that invokes particles travelling faster than light; they are called tachyons. However as things stand at the moment such theories are either simply wrong, or else they are using the word 'tachyon' in a rather subtle technical sense, as a way of calculating parts of a process which overall does respect the normal speed limit.
This speed limit is not so much a limit on speed as a statement about the structure of the world as a function of both space and time.
Finally, you might like to note that this question is always coming up and there are plenty of other similar questions and answers already on this site. Perhaps I should have directed you to one of those and then added this answer there. Anyway, there it is.