We establish that the effect of intensified deposit market competition, measured by reduced switching costs, on the probability of bank failures depends critically on whether we focus on competition with established customer relationships or competition for the formation of such relationships. With inherited customer relationships, intensified competition due to lower switching costs destabilizes the banking market, whereas it stabilizes the market if we focus on competition for the formation of customer relationships. We characterize the factors important for evaluating the effects of intensified competition on stability in a market with unattached as well as locked-in depositors.