More than once.
When, in Noel Ignatiev's words, did the Irish become white? My mother (descended from a long line of Boston WASPS) still uses "paddy wagon" idiomatically, unselfconsciously, and unreflexively for police prisoner transport van--a linguistic relic of a time before the Irish were white.
And here in California (at least) Asian-Americans (those whose ancestors come from East Asia *and* from South Asia, with the exception of those whose ancestors come from the Philippines and from southeast Asia), are rapidly becoming becoming white (or maybe in world-historical context it is us European-Americans who are becoming honorary members of the Central Country).
Brad DeLong