My 18 month old kitten has been pulling open cabinet doors and the door to my sunroom since I first got him at 8 weeks old. He learned it pretty quickly from watching my other cat pull open the bathroom door, who was 18 months old at the time. I think my kitten might have a knack for opening things though, because his technique now includes stretching up to the doorknob, putting one paw on each side and turning it to unlatch the door before he tries pulling or pushing.
Some cats just don't want to pull open doors. In my experience, it's the most curious ones, the ones who can't stand to have any door in the house closed because they want to know what's on the other side, who bother to try and figure out doors. Some cats are too lazy and couldn't care less as long as there is a warm spot available.
EDIT: Haha, just as I finished typing this, I heard the doors to the cabinet under my bathroom sink rattle. When I went to check it out, I found my kitten playing around inside the cabinet, with the door closed behind him (for anyone reading this, don't worry, I don't keep anything toxic down there).