experimental physics - Does mathematical sloppiness in quantum mechanics ever produce incorrect predictions?

Does mathematical sloppiness in standard quantum mechanics ever produce predictions that don't pan out? I'm not talking about things like the WKB approximation, but instead subtle functional analytic issues, such as assuming every Hamiltonian is self-adjoint, has an eigenbasis of bound states, domain issues, etc. I don't know of any such experiment, but it is conceivable that a bad enough physical Hamiltonian exists so that the standard methods fail.

I emphasize that I am looking for actual experiments people have done, not thought experiments and not contrived counterexamples.