I didn't necissarily switch accents...
But I spent about two years in London for work and in about the first three weeks I started having difficulty pronouncing words. I'd trip over my tongue often and started chopping off syllables. It ended after a while but for some time I had to consciously focus on enunciation to get anything out.
When I came back home for Christmas to my family my mother noted that I had dropped the harsh 'a' sound and was saying 'ah' instead. But other than that I picked up my old Washingtonian accent immediately back in America.
My accent was definitely
effected by exposure to British accents but it was more of a watering down than anything else. I'll never get a pure British accent (and I'm not trying to) and British people can still always tell that I'm an American.
I don't know whether or not you're talking about native or non-native English speakers though. Because you can learn BritE and learn AmE and transfer between the two as well. It just depends I suppose.