Type II and quasi-type II nanocrystals with thick shells exhibit reduced blinking. However, after a number of monolayers, the influence of the shell thickness is found to vanish. Using a two-band Kane Hamiltonian, it is shown that this behavior is a consequence of interband coupling and asymmetric confinement of electrons and holes. Interface alloying provides an additional, order-of-magnitude contribution to the Auger suppression, in agreement with recent experiments. The existence is predicted of critical shell thicknesses that strongly quench Auger processes for any core size.