Suppose we have a Hamiltonian that depends on various real parameters. When tuning the values of these parameters, the energy eigenvalues will often avoid crossing each other. Why?
Consider what happens if there is a crossing. A crossing would imply a degeneracy in the system. A degeneracy would imply a symmetry. It would be unnatural for a perturbation to introduce a symmetry into a system, and so the eigenvalues cannot cross generically, but can under special circumstances.