M.C.
Oh yes, hair can definitely do this. I have a hair trap over my drain, and I *still* have to get into the drain and pull up wads of hair every few months to keep it draining smoothly.
I would reccomend you do the same... Get a cheap drain cover to block most of the hair from going down the drain, then every few months, just remove the drain cover (or the center stopper thing, if that's the type you have...) and run a snake down it. Or, if you don't want to go to that trouble, there is a chemical drain cleaner that is specifically formulated to dissolve hair clogs. I hate using chemicals in my drains, but it might work for you.