When it interferes with other things in your schedule that you should've prioritized higher.
Well if we're talking about this kind of situation, the above obviously applies.
However if you put it into a much wider scale (ie fandom) it becomes a lot more difficult to distinguish I think. There are certain people I've spoken to on the Bioware forums who take their love of Mass Effect/Dragon Age very very very far. I'm sure that's the case in most fandoms and, at the risk of being flamed, I think a lot of that is because it's got a higher percentage of female fans than other game series. And women, more than men in my experience, tend to get into every nook and cranny of a game's fandom.

But it's all subjective. When does being a supporter of something, say a game, become obsessive? When you're playing in
all your free time? When you start writing fanfiction about the game? When you follow every single voice actor on Twitter and tweet them regularly? When you use your annual leave from work to go to a single convention that showcases the game?
Or is it only when you do all of those things combined? I guess any one of those is probably considered fairly normal but when someone does all of it....it's more than a little obsessive.