We study a monetary, general equilibrium economy in which banks exist because they provide intertemporal insurance to risk-averse depositors. A "banking crisis" is defined as a case in which banks exhaust their reserve assets. Under different model specifications, the banking industry is either a monopoly bank or a competitive banking industry. If the nominal rate of interest (rate of inflation) is below (above) some threshold, a monopolistic banking system will always result in a higher (lower) crisis probability. Thus, the relative crisis probabilities under the two banking systems cannot be determined independently of the conduct of monetary policy. We further show that the probability of a "costly banking crisis" is always higher under competition than under monopoly. However, this apparent advantage of the monopoly bank is due strictly to the fact that it provides relatively less valuable intertemporal insurance. These theoretical results suggest that banking system structure may matter for financial stability.