For many of the obvious ways to pose the question, atomic theory is already false - multiparticle states are the real building blocks, and you can do pretty unusual things with them if you try hard enough. I think the most sensible thing to ask about is sudden failure of properties that we normally ascribe to atomic theory, like ratios working in chemical reactions or quantum mechanics predicting NMR spectra. In which case, I'd need said failures to replicate to be as good as the supporting evidence, or propose a simple-ish mechanism, like "we're in a simulation and it's just changed." Taking that as my lowest standard, I'd be satisfied with a standard star-blinking pattern, or maybe a pi-message, the usual sort of thing, with null probability with an exponent somewhere in the -25 range.