Neutrino experiments are a good example! A mistake on my part. I think you're right that the CKM parameters have been bounded more tightly with non-accelerator based neutrino experiments. (Of course, that's still HEP, and it still gets plenty of funding, so it doesn't detract from my point.)

On the other hand, superconductivity is not a good example. It's a still-confusing phenomenon, but the vast majority of physicists believe to follows directly from the known laws of non-relativistic quantum mechanics.