# Non-degenerate or degenerate perturbation theory for a non-degenerate level of a system with other levels degenerate?

To decide whether I have to use non-degenerate or degenerate perturbation theory, I have to look only on whether the energy level I am calculating corrections to is degenerate, the degree of degeneracy of the other levels is always immaterial for this, correct?

For instance, if I calculate corrections to ground state $n=1$ of the hydrogen atom, I use non-degenerate perturbation theory, but for all other levels $n\geq 2$ I use degenerate perturbation theory and might get splitting of energy levels, depending on the perturbation (e.g. the Stark effect of hydrogen atom).

The motivation behind this question is that although it seems plausible, in textbook chapters on non-degenerate perturbation theory it is often assumed that no eigenvalue of the unperturbed Hamiltonian is degenerate. Maybe there is something more behind this I overlook?