This may be a temporary problem, but it has been confirmed by several people, including Giz and Gizzettes: right now, you can't set your date of birth as the 31st of any month on Facebook.
The social networking site thinks that January, March, May, July, August, October, and December only have 30 days. As far as we can tell, there's no solution for this. So, if you are one of the five people who has never signed up for a Facebook account and were born on the 31st of any of those months, you are fresh out of luck. [Thanks Anna Holmes!]