Using micro- and nano-scale resonantly excited PL and PLE, we study the excitonic structure of CdSe/ZnSe and CdTe/ZnTe self assembled quantum dots (SAQD). Strong resonantly enhanced PL is seen at one to four optic phonon energies below the laser excitation energy. The maximum enhancement is not just one phonon energy above the peak energy distribution of QDs, but rather is 50 meV (for CdSe dots) or 100 meV (for CdTe) above the peak distribution. We interpret this unusual result as from double resonances associated with excited state to ground state energies being commensurate with LO phonons. Such a situation appears to occur only for the high-energy quantum dots.