We analyze the conditions of emergence of a twin banking and sovereign debt crisis within a monetary union in which: (i) the central bank is not allowed to provide direct financial support to stressed member states or to play the role of lender of last resort in sovereign bond markets, and (ii) the responsibility of fighting against large scale bank runs, ascribed to domestic governments, is ensured through the implementation of a financial safety net (banking regulation and government deposit guarantee). We show that this broad institutional architecture, typical of the Eurozone at the onset of the financial crisis, is not always able to prevent the occurrence of a twin banking and sovereign debt crisis triggered by pessimistic investors' expectations. Without significant backstop by the central bank, the financial safety net may actually aggravate, instead of improve, the financial situation of banks and of the government.