If it can't be falsified, it isn't a proper theory. Sting "theory" falls into this catagory, since right now it can't be proven or disproven.
Yes, big bang theory can be proven to be false, if some new peice of evidence came up that totally contradicted it. You can't really definitively prove a theory, since there is always something else that you could find that disagrees with it. But some theories, like special relativity, can be pretty much proven to be correct.