The EPR (Einstein Podolsky Rosen) paradox was, when created, a thought experiment. John Bell showed that it was possible to test whether 'hidden variables' could be a resolution to this paradox. Experiments were done, on the Bell inequality, and the results are pretty darn unambiguous - the universe is a very, very strange place.
I have started this thread so we have a single place in BAUT where we can discuss this topic. It is a topic which is, it seems, widely misunderstood (e.g. this heusdens post seems to imply a hidden variable solution is possible). It is also a topic with relevance to those who come at science with a classical view (e.g. ErdExpMann and ColdCreation - I'll add links later).
There are many excellent explanations of this topic on the web, and several places where you can discuss it, to whatever technical or philosophical depth you are comfortable with. If there is sufficient interest, I will add links to some of these.