I have some sort of accent (I moved to the U.S at 10 after living in Germany for 4 years, Denmark in 6), but when I am in America, most Americans (maybe 99.9999% of them) think I'm from the U.S - although generally from some other part of the U.S than the one they're from.
I did have the experience of one guy thinking I was from Germany after 20 years in the U.S (hence the 99.+ percentile) but he was quite the outlier.
However when Americans meet me in Europe they immediately identify me as having a non-American accent, which really messes with all my European friends since they think of me as being essentially American.
So this disability thing seems quite context dependent.