I get the ethnicity question all the time from my students. In appearance I am ethnically ambiguous (not surprising, since I am mixed race). I get everything from Latina to Native American to East Asian to Eastern European as guesses (I am East Asian/WASP). My parents also gave me an ethnically ambiguous first name. But I have a very WASP last name. Many people assume that my last name is my married name (which I find really funny, since I'm not married), and that my middle name is my maiden name (it's my mother's maiden name).
I probably have somewhat compounded the issue, since I'm in an MLA field, and my languages of specialization are not languages that at all "match" my appearance -- they're just languages I fell in love with.
Some of my students just flat out ask me, which doesn't bother me. I am highly amused by those who sneakily try to figure it out by asking indirect questions.
I hear you peppergal... I've seen/heard versions of what you've experienced on my campus. But, check this one out:
I teach many courses dealing with a European country that is widely stereotyped in the States as a "hyper-Aryan" (blond hair, blue eyes, you name it) country. Whenever I teach a course on said country, I inevitably have at least one student who asks me what part of that country I'm from and how I learned to speak English "so good."
I am unambiguously of East Asian descent and am a native speaker of American English.
While it is not *impossible* to have a person of East Asian descent be from that country and also be a native speaker of American English, it is, well, *highly improbable*--at least by most stretches of the cultural imagination (and of imagination, my students appear to have much...good for them, strange for me).