I propose a dynamic general equilibrium model in which strategic interactions between banks and depositors may lead to endogenous bank fragility and slow recovery from crises. When banks' investment decisions are not contractible, depositors form expectations about bank risk-taking and demand a return on deposits according to their risk. This creates strategic complementarities and possibly multiple equilibria: in response to an increase in funding costs, banks may optimally choose to pursue risky portfolios that undermine their solvency prospects. In a bad equilibrium, high funding costs hinder the accumulation of bank net worth, leading to a persistent drop in investment and output. I bring the model to bear on the European sovereign debt crisis, in the course of which under-capitalized banks in default-risky countries experienced an increase in funding costs and raised their holdings of domestic government debt. The model is quantified using Portuguese data and accounts for macroeconomic dynamics in Portugal in 2010-2016. Policy interventions face a trade-off between alleviating banks' funding conditions and strengthening risk-taking incentives. Liquidity provision to banks may eliminate the good equilibrium when not targeted. Targeted interventions have the capacity to eliminate adverse equilibria.