I think it's absolutely possible; I'm a natural extrovert who periodically suffers from social anxiety. As a kid I was always told I was an introvert, but as my withdrawal from social situations back then had to do with factors completely out of my own control, I eventually came to the conclusion that the label was at best a bad estimation. The only scientific gauge I've submitted myself to in this regard is the Myers-Briggs test, but even there on some tests I get the result of INTJ but I more often test as ENTJ.

It's not really an either/or situation, actually, extraversion and intraversion. Everyone has elements of both; it's just a question of which is more dominant, and to what degree.

There's also a lesser known third category called "ambiversion", to describe the qualities of someone who is comfortable in social interaction but also enjoys time alone. Something tells me it might be more productive though (and accurate) to consider the ways in which one enjoys or experiences extraversion and intraversion, than to label oneself one or the other, although for someone who dominantly experiences only one (or feels they experience them exactly equally) it might make more sense.