This may be a temporary problem, but it has been confirmed by several people including ourselves: as of this moment, you can't set your day of birth to the 31st of any month in Facebook.
The social network 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 was born on the 31st of any of those months, you are out of luck. [Thanks Anna Holmes!]