For me, the only real problem I have with any particular anime is when the story itself bores me. I treat anime much the same as I treat any other TV show or film, whether it's from Japan or not. I won't watch a US or UK drama series that I find to have a crap story, likewise if I don't enjoy the story of a particular anime then I simply won't watch it. Other than that, I do get somewhat annoyed, yet amused, at the people who take anime far too seriously. Though I would never call someone out on it, I can't help but laugh at the people who go overboard with their devotion to an anime. I mean, fine, I'll admit I do ship characters and stuff in my head, and have ideas for fan fiction (though I've not got around to writing anything yet...). But going overboard on something is just silly. Though, then again, I still laugh anyway at people who do that with anything else. Don't even get me started on Trekkies...
Oh, and I've never understood why some people refuse to watch particular anime in English when dubbed, because they see the voice acting as being bad. I mean, whenever I watch anime I don't focus on the voice acting, I just listen to what's being said. I'm watching Higurashi No Naku Koro Ni at the moment and the voice acting sounds fine to me, if a little over the top at times, but most comments I see online are that the voice acting is crap and that subs are much better. Anyone care to enlighten me here? After all, if I wanted to read subtitles I'd just go and read the manga instead of watching an anime...