I may have heard this or understood this incorrectly, so if i am asking the wrong question, feel free to correct me. As I understand it, if you have a degenerate set of simultaneous eigenvectors, you haven't specified a complete set of operators. For example, the hydrogen atom. You typically have the Hamiltonian, angular momentum, spin in elementary discussions. However, as I understand it, the eigenfunctions aren't complete and a final operator is needed (I forget the name). The thing I don't understand is a) how you would determine that final operator and b) how exactly adding operators would even help to prevent a set of eigenvectors from being degenerate.