Einstein, in his book, RELATIVITY, written for the layman, shows the necessity of this using a THOUGHT EXPERIMENT based on the Principle of Equivalence. It's very interesting, especially because you actually get a quantitative result from it.
I note Galileo also proposed a THOUGHT EXPERIMENT to show that objects must fall ( or accelerate ) at the same rate in a gravitiational field. He supposed an iron ball, say, to be cut in half and the two halves to be regarded as separate objects falling side by side. Then repeat the experiment with them touching. "In what way is this pair different than the original ball?" he asks.
MYTHBUSTERS did an experiment, which was purely kinematical. That is, the necessary result could be seen by conducting the experiment in thought. The "myth" being tested was that a ball shot backward from a moving vehicle with a muzzle velocity equal and opposite to the vehicle would fall straight to the ground.
Of course, there could be questions of air turbulence and such in the execution, but they did get a confirming result, and Adam was rather excited about, too.
Einsteins thought experiments involving light assume that the Michelson/Morley experiment actually did fail and the failure was not simply a case of not having good enough equipment; nonetheless the experiments were run continuously in the 1920s and 30s at higher elevations and with significantly better equipment, and they did not fail. Try google searches on "Dayton Miller" for that one.