The notion that flexible exchange rates insulate a country from foreign shocks is well grounded in theory, from the classics (Meade, 1951; Friedman 1953), to the more recent open economy literature (Obstfeld and Rogoff, 2000). We confront it with new evidence from Europe. Specifically, we study how shocks that originate in the euro area spill over to its neighboring countries. We exploit the variation of the exchange rate regime across time and countries to assess whether the regime alters the spillovers: it does not - flexible exchange rates fail to provide insulation against euro area shocks. This result is robust across a number of specifications and holds up once we control for global financial conditions. We show that the workhorse open-economy model can account for the lack of insulation under a float, assuming that central banks respond to headline consumer price inflation. However, it remains puzzling that policy makers are ready to forego stabilization of economic activity to the extent we found in the data.