The term is misleading as often the observed degeneracy is not accidental at all, and is a consequence of a 'hidden' symmetry which is not immediately obvious from the Hamiltonian in a given basis. The non relativistic Hydrogen atom is a good example of this - by construction, its Hamiltonian is invariant under the full rotation group in 3 dimensions, SO(3). A less obvious feature is that the Hamiltonian is also invariant under SO(4), the extension of SO(3) to 4D, of which SO(3) is a subgroup (another way of saying this is that all possible rotations in 3D are also possible in 4D - we just don't rotate about the additional axis). This gives rise to the 'accidental' degeneracy observed in the Hydrogenic eigenspectrum.