[sidek]

The problem (not having a condensed matter background, so my answer might not be entirely right) is that we can't solve the Schrödinger equation exactly for many-atom systems.

So we need to approximate our solutions to be able to get a good idea of what properties a material will have. Our approximations are pretty good and we have a decent idea of when some things superconductor. But our theories of superconduction aren't yet perfect, and sometimes the approximations can fail//generate behavior unexpected by our theories.

[selimthegrim]

In general, we can only predict ground state properties with accuracy. Properly predicting high temperature superconductivity requires generating information about the Fermi surface that is not just a matter of ground states. If this is standard BCS model stuff then you might be able to get away with some rules of thumb.