It is by now generally accepted that foreign exchange returns exhibit “heavy tails” as measured by the so-called tail index. However, it is unclear whether the tail behavior remains stationary in the presence of recurrent switches in the exchange rate regime. We therefore test the null hypothesis of tail index constancy by applying a single breaks test “in rounds” which enables the detection of multiple breakpoints. We are able to identify multiple jumps in the tail index of currency returns. Moreover, some breaks coincide with documented shifts in monetary and exchange rate policies.