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When a country's banking system becomes more linked to the global banking network, does that system get more or less prone to a banking crisis? Using model simulations and econometric estimates based on a world-wide dataset, we find an M-shaped relationship between financial stability of a country's banking sector and its interconnectedness. In particular, for banking sectors that are not very connected to the global banking network, increases in interconnectedness are associated with a reduced probability of a banking crisis. Once interconnectedness reaches a certain value, further increases in interconnectedness can increase the probability of a banking crisis. Our findings suggest that it may be beneficial for policies to support greater interlinkages for less connected banking systems, but after a certain point the advantages of increased interconnectedness become less clear.