We study information acquisition and dynamic withdrawal decisions when a spreading rumor exposes a solvent bank to a run. Uncertainty about the bank's liquidity and potential failure motivates depositors who hear the rumor to acquire additional noisy signals. Depositors with less informative signals may wait before gradually running on the bank, leading to an endogenous aggregate withdrawal speed and bank survival time. Private information acquisition about liquidity can subject solvent‐but‐illiquid banks to runs, and shorten the survival time of failing banks. Public provision of solvency information can mitigate runs by indirectly crowding‐out individual depositors' effort to acquire liquidity information.