How do we know that experimental evidence of special relativity can't be attributed to other, unrelated effects? While there are many experimental results that seem to align with the predictions of special relativity--some examples being muons from the upper atmosphere reaching the Earth's surface despite their short half lives as well as atomic clocks being flown in jets around the Earth reading different times than identical clocks left on the ground after the flight (i.e. Hafele-Keating experiment). My question is, how do we know that the results of these experiments (i.e. their disagreement with classical mechanics) are direct results of (and thus direct verifications) of special relativity rather than being caused by other unrelated effects such as decay rates varying with altitude and other external properties such as atmospheric pressure, or even with varying amounts of gravity (these are just some examples of ideas of other possible explanations that I had)? In addition, how do we know that these experimental variances from classical mechanics are caused by relative motion rather than gravity?
Essentially, how so we know that experimental results that seem to show special relativistic effects--such as time dilation or length contraction--due to relative motion can truly be directly attributed to special relativity?
Do these experiments truly show (i.e. provide evidence for) the existence of special relativity or do they simply give the illusion of its existence due to other unrelated effects that give similar results?
 A: The primary justification for special relativity is not that muons last longer as they fall through the atomosphere at high speed. That is just an implication to check the otherwise well confirmed theory. Starting from electromagnetic theory - which is backed up by pretty much every piece of electro-mechanical equipment (staying away from quantum effects) we get an invariant speed of light. From this point we have no choice but to follow something like Einstein's approach and end up with special relativity. General relativity is not quite so clear - but is close to assuming that the world is locally rather than globally special relativistic.
Other experiments included measuring the rates of clocks at the top of buildings and in aircraft, and gravity probes A and B (NASA satellites). There is much theoretical and experimental evidence that is ubiquitous in modern science and technology, that goes well beyond the comment that muons last longer at high speed.
A: One way to approach the problem of alternate explanations is as follows. 
Any alternative "model" we might cook up as a replacement to special relativity would need to be mathematically structured in just such a way as to furnish precisely the same experimental effects as predicted by special relativity in all other experimental contexts, while at the same time being mathematically distinct from special relativity. At the same time, it must neither screw up other well-understood parts of physics in the process nor require the invention of some unknown and heretofore unobserved physical effects to make the new model work. This places extremely tight constraints on the mathematics of any and all alternatives to special relativity. 
A: In addition to the other answer, all of particle physics data  ( a huge number) validate special relativity and all new predictions of particle theory , the standard model, are continuously validated, no falsification has been seen.
It is possible that a future theory of everything  may have a more complicated mathematics, but it should at the overlapping phase space reduce to special relativity, because it should explain the same data. Actually General relativity gives a different mathematical description than the Lorentz transformations, that is why the GPS measurements are corrected for both general and special relativity effects, but for flat spaces, the mathematics reduces to Lorentz transformations.
A: Each theory has a domain of validity. A proper scientific reasoning would follow those lines: 
1) does an experiment or observation expose a domain where special relativity is not valid? 
2) if so, then for all phenomena described by special relativity in that domain, we need to find another framework/model. 
The question goes like this: 
3) for all domains where special relativity is effective, can we replace it by another model? 
Well of course we can. But what's the point? We would have to replace a single unifying framework by a collection of domain-specific models, and do this with the (huge) constraint to have these models behave consistently where their domains boundaries meet. 
And where do you stop? You can apply this reasoning to the whole of physics and attempt to get rid of all deep, unifying theories and replace them with a myriad of epicycles-like models based on whatever core ideas you want to keep as valid (so first you would have to explicit what these ideas are). It would be a huge endeavour, that would have to be done again for each experiment that, thanks for example to a better precision, would invalidate any one of these models. 
In the process we would also loose all insights we have about the nature of the world behind what we can immediately perceive. Instead we would have to work endlessly to enforce an arbitrary worldview based on the set of core ideas considered by some authority as valid. 
It's essentially doing science backwards.
