Have any known experiments ruled out travelling faster than the speed of light?
Or is this just a widely accepted theory?
I think the comments are explaining it just right...
No, there isn't. Similar to the statement that "there's no such thing as 100% efficiency", there doesn't exist an absolute proof to anything (especially in Physics)
No again. The speed of light and the particles traveling closer to speed of light are experimentally verified by the consequences like time dilation (for e.g. Muons). But, there are a lot of theories regarding the FTL because physicists showed a lot of interest back at 1950s on the subject "Tachyons"...
SR is accepted widely because of the verifications it provided and the predictions it made. And for this reason, SR is most widely accepted all around. In fact, it's more useful in particle physics when accelerating particles to high energies.
This needs to be emphasized :
No theory, and Special Relativity that is used in physics to describe data is a theory, can be proven, in the way a mathematical theory can be proven with a QED at the end.Theories can only be validated by the agreement with the data on the predictions they make.
That is why the OPERA faster than light neutrinos (wrong) observation gripped our attention.
Even one falsification of a theory's predictions invalidates a theory and a new axiomatic formulation, a new theory, would have to be found .
There has been no falsification by experiment of special relativity.
That is the difference between physics and mathematics.
Yes, there is definitive, mathematical proof within the framework of general relativity that a mass cannot be accelerated to or beyond the speed of light...assuming relativity is correct and complete.
The next question is "Is relativity a correct and complete description of the universe?" No, it isn't. But it's pretty good, and it doesn't look like we're going to find a way around that particular speed limit.