The King of Thailand is a natural-born US citizen, and his accent blows. There are plenty of people out there who were born in the US who have lived all but the first few days/weeks/months of their lives outside the US. So, judging people on accent is pretty stupid.

USVI is the corner case, since people can swim over from the British Virgin Islands. I remember a long discussion a bunch of us had on that last year. The OP can look up that thread and enjoy the debate.