I had a similar experience to your cousin. I grew up in Maryland, just outside DC and thought myself accentless although I can definitely pick up an accent when I listen to people from that area now–even my relatives. Still I went away to school and roomed with a guy from Brooklyn. I apparently picked up enough of the Brooklynese that for several years thereafter people would ask me if I was from Queens because they knew it wasn’t quite Brooklyn but it had to be somewhere in New Yawk.
But then I moved on to North Carolina and after 4 years of talking to mostly poorer, rural North Carolinians I definitely had a drawl.
Now, after living in the far west for several decades, I find that my accent adapts. I drawl when in the south, a revert to the Brooklynese when in the northeast.
Many years ago there was a fascinating 10-hour series on PBS by Robert MacNiel called "The Story of English" ( http://www.imdb.com/title/tt0198245/ ). I taped it off the tube with my first-generation VCR and have watched the less-than perfect recording many times. I’d love to see a reasonably priced, cleaned up version come out on DVD because it was/is a wonderful program about speech patterns, language use and accents. Among the fascinating discussions are the Elizabethan English spoken by residents of Tangier Island in Chesapeake Bay until very recent times (because the place was settled in the 1600’s and has been pretty isolated since) and the Gullah speach of the Georgia Sea Islands. One program talked about the residual "Scottishness" of the Southern "hillbillies" whose ancestors were, for the most part, Scotts-Irish immigrants. And there’s so much more. It’s a great series if you feel like buying it or get a xchance to see it somewhere.