I usually get that bored/annoyed feeling whenever I sense someone is fishing for a specific sort of reaction from me. I've written on here before (on @SilkRoad 's thread about genuine self-esteem vs. arrogance) about people who are narcissistic and interact with others mainly to get admiration; when it seems like someone is doing that I find it dull, irritating and sort of invasive (since they've already decided what my opinion, thoughts and feelings are supposed to be). When someone is being a drama queen and is clearly trying to get others to act as a source of pity, sympathy, or fascination I feel similarly. In each case, they've decided beforehand how I'm supposed to respond and are trying to stage manage me into it. I hate that.
(also noticed people are experiencing drama queen/king types as "energy vampires", people who "drain" you....I've always found narcissism or pushing for admiration similarly tiring)
Another thing this brought to mind: I read a book called 'Lip Service', which discussed the ways in which women are expected to be superior to men (more virtuous, more nurturing, better at relationships and intimacy, etc.), and the effect these expectations have on women. One thing the author discussed is the idea that self-disclosure, talking about your vulnerable feelings, is the hallmark of intimacy, and women are supposed to be good at this. So talking about their problems, sometimes dramatizing them and making them seem worse than they are, often becomes a central feature in women's friendships. It's not that sharing your problems is a bad thing; it can be wonderful. But if it's the main way friends feel close to each other, when "feeling bad is fused with feeling connected", there can be problems. She mentioned that this sort of friendship could often use other ways of feeling close, some of which SilkRoad mentioned: shared activity, interesting discussion, etc.
I don't see anything wrong with being bored by it. It is boring. *shrug*