Are users reluctant to give their complete date of birth?

I am required to ask users their DoB, but only for verification purposes, to make sure the user is over 18 years old. This info is one of the 6 things I ask in a Sign Up form to participate in a beta site, so it's a small form.

Because I am slightly reluctant to give a precise date myself, I wonder how necessary the info really is. I would be more comfortable myself picking a month and a year, but no so much a day (security reasons.) However I haven't seen this anywhere, and it feels quite strange.

In short:

Would it make sense / feel okay to ask for MM/YY only instead of DD/MM/YY?