I guess it really depends on the subject so let me elaborate...
Which is worse, overestimating or underestimating your own abilities/intellect?
Each have their merits but I (personally) think that underestimating your own abilities is a much worse offense than overestimating them. With the former, you are doing yourself a disservice by not living up to your full potential or at least not fully realizing your potential. There are probably several reasons why people tend to underestimate their abilities:
1.) because they've been repeatedly told that they're NOT good enough,
2.) because they've never been given credit for their accomplishments, and
3.) because they were never given a chance to shine (among others).
I'm one of the guilty ones (who tend to underestimate) although I'm not really sure why I do it. I guess partly because we were often told not to boast when we were younger. Arrogance is not very ladylike (not that I was ever arrogant, at least I don't think) or so they say. Which I guess is partly cultural as well. Growing up in the Philippines, in a strict and conservative Catholic environment, we were always taught to be humble and modest and polite. Fast-forward a few years later to living in the U.S. where individuality and being outspoken is highly valued, one gets into a state of identity confusion (especially since I moved to the US during my adolescent years).
Another reason why I slightly undermine my abilities is, partly, because I tend to be quite timid. I'd rather be on the sidelines than in the limelight (which is paradoxical in and of itself but such is life). The limelight doesn't scare me. It's the responsibilities that come with it that I'd rather not bear.
I recently realized that by underestimating myself I am only holding myself back from experiencing everything that life has to offer. I owe it to myself to be the best I can be and reach for the stars. As Eleanor Roosevelt once said, "The purpose of life is to live it, to taste experience to the utmost, to reach out eagerly and without fear for newer and richer experience."