I did.
Well, I've still got an "obsession," which would be music, but it's technically not an obsession, just a major part of me. I think about other things, and I do other things, and I talk about other things, but it's kinda like it's constantly running in the background, it's who I am. But it's not on my mind constantly.
As for "periodic obsessions:" I don't feel there is anything wrong with those. Right now it's Stargate. But it's like a book that you're really into. You just keep jumping endlessly from each chapter to the next, and spending a little time on the side speculating what you think might happen or stuff like that. I've been watching SG-1 constantly lately for the past couple weeks on Hulu, it's quite addicting for me as I like the concept of the storylines. But the periodic thing is another something else I would not label as "obsession."
So I guess you could grow out of obsessions. I've grown out of a lot of other traits as well. I can rarely watch a movie more than once unless it's extremely funny enough it keeps its comic value, or the storyline is possibly epic enough.