Yep, it's entirely possible to develop asthma at any age.
I strongly suspect I have mild asthma. I was never able to keep up with everyone else in gym class because of my breathing issues, and I'd be out of breath for hours afterward - I could have gym in the morning and not have my breathing return to normal until an hour before I went home. It was particularly bad as a teen - I remember I'd sometimes be lying on my bed struggling to breathe and wondering if that would be the time I'd end up in the hospital (but I never did, it always ended up settling down on its own). I don't like going to the doctor because I'm always afraid they won't believe me or won't find anything wrong even when I feel like something's wrong. This is actually exactly what happened when I finally worked up the courage to mention it the first time - the doctor said my lungs sounded fine (yeah, and I wasn't having any trouble breathing at the time, either ) and was hung up on the fact that I didn't cough when I'd have breathing trouble. I've since realized that I unknowingly conditioned myself not to cough since I was a kid, because it makes noise and I hate drawing any attention to myself. I do get a sensation like I want to cough, but deeper in my chest than usual - it feels like I want to cough, but it won't actually help. I have a different doctor now, and a few months ago, she put me on Singulair for my allergies, and it didn't really help with that, but it did help my breathing - it didn't completely fix it, but it's better than it was. I could even do the first week of C25K and not be out of breath for an hour afterward (although week 2 was another story) - I thought I'd try it since my lungs felt so much better. The fact that that helped makes me more confident that it actually is asthma. I'd like to be officially diagnosed, but my medical insurance doesn't cover doctor visits anymore, so my parents really don't want to have to take me unless it's absolutely necessary.
_________________
Yet in my new wildness and freedom I almost welcome the bitterness of alienage. For although nepenthe has calmed me, I know always that I am an outsider; a stranger in this century and among those who are still men.
-H. P. Lovecraft, "The Outsider"