One particularly fascinating example of this I have found is the following. The delta function potential has no effect in nonrelativistic quantum mechanics in spatial dimensions greater than or equal to 4. This was first proven here: https://www.sciencedirect.com/science/article/pii/002212367290033X.
The QFT analog is the result that the $\phi^4$ theory is trivial in spacetime dimensions $d>4$. This was proven here: https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.47.1.
The reason I call this analogous to the nonrelativistic result is that if you construct a (nonrelativistic) many body quantum mechanics system with a delta function potential interaction, this will give you an interaction term equivalent to the $\phi^4$ in the QFT.
My question is, is it reasonable for the nonrelativistic result to be, at least suggestive of an analogous QFT result?
Now, the stability of the hydrogen atom in spatial dimensions higher than 3 is, as far as I am aware, an open question. I guess the reason for this question is my wanting for the answer to the stability of the hydrogen atom to have any suggestive power to nonperturbative results of QED in high spacetime dimensions.