Is it possible to test if people are rational? I think the answer, practically, is a very short no: if a rational person tries to achieve his most preferred outcome of the ones that are available, we can't distinguish the rationality or irrationality of his choice from his preferences. That is, if I don't know what you like, I can't tell if you did something because you liked it or because you're "irrational".
Yet rivers of ink have been spilled trying to "prove" or "disprove" models of rational choice. The most famous study of the type is the "Allais paradox", discussed here. It says that when you pose different choices to people, their responses to pairs of choices are "inconsistent" with each other because the two choices really represented the same cash outcomes.
Whether you look at the question being asked by this type of work "are people rational?", or "what do people care about?", it's pretty clear that any observation cannot answer either of these without knowledge of the other. In "The Methodology of Positive Economics" (pdf) Milton Friedman made the valid, general point that
"If there is one hypothesis that is consistent with the available evidence, there are always an infinite number that are."
It just so happens that if we interpret some piece of evidence as being consistent with "people are irrational", one of the "infinite number" towers above all others: "you guessed the preferences wrong". There's nothing wrong with trying to figure out how to better model the decisions of people, but claiming to have proved irrationality is nonsensical.