It is shown that in those autoassociative memories that learn by storing multiple patterns of activity on their recurrent collateral connections, there is a fundamental conflict between dynamical stability and storage capacity. It is then found that the network can nevertheless retrieve many different memory patterns, as predicted by nondynamical analyses, if its firing is regulated by inhibition that is sufficiently multiplicative in nature. Simulations of a model network with integrate-and-fire units confirm that this is a realistic solution to the conflict. The simulations also confirm the earlier analytical result that cued-elicited memory retrieval, which follows an exponential time course, occurs in a time linearly related to the time constant for synaptic conductance inactivation and relatively independent of neuronal time constants and firing levels.