We know that on the surface of the Earth, there is a maximum distance you can get from any other object. We could have confirmed this empirically by placing transmitters at points around the world and seeing how far away they were, but we instead we proved it conclusively by showing that the Earth is a sphere, and given the way that distances work, it doesn't make sense for something to be more than 20,000 km away along Earth's surface. If you try to, the object will seem to appear closer behind you, though from the outside we know it just moved across the point opposite you on the Earth's surface.

There's an analogous fact about spacetime. Minkowski space, which seems to be what we live in, does not geometrically allow for something to accelerate above the speed of light. It's not simply something we haven't observed, it's more like trying to get more than 20,000 km away on Earth's surface - the way velocities work in this geometry make that physical action kind of nonsensical. We 'proved' things can't go faster than light by experimentally confirming various effects that we would expect in a Minkowski space like length contraction and time dilation.

It may be that things can go faster than light, but if so it will require new physics, and will mean we don't really live in a Minkowski space, but something that usually behaves similarly but something is different. This is very probably the case, since we know that general relativity is an incomplete theory, but so far no extensions allow for faster-than-light travel either.