Personally I can skip pretty much anything, but being bothered - or even able to - depends on whether I'm watching it on TV, the computer, or some other medium, and if it's easy or even possible to skip the thing.
A lot of my watching of actual series of scripted shows or cartoons recently has been with DVD box sets, and for most of them the makers have been kind enough to put a chapter mark between the end of the intro and the start of the actual content (and in some cases you can skip the recap too), though in one particular case I did end up having to get quite familiar with my DVD player's "seek by time" function, and the 8x fast forward for another.
If it's on TV, or streaming media, usually either you can't skip (naturally), or it's so unreliable or prompts such a loading delay it's just not worth it... maybe I'll go grab a drink instead. VHS or VCD, well, YMMV depending on the hardware. My own isn't too bad for it...
However...
SomeLameStuff said:
I can't skip the opening to Megas XLR, I just dig giant robots too much!
Popadoo said:
Actual intros I can't skip:
...ones like those, I tend to let run (though I'm yet to actually watch much of either show 9_9). Notice the connection? Yep, the theme tunes are either pretty mainstream in terms of music, or missing altogether, and they're less than a minute long - in the case of Megas XLR, barely more than 30 seconds (similarly, the powerpuff girls and other such short-episode Cartoon Network things)
Even my very favourites - works of brilliance like Cowboy Bebop, or nostalgia bombs like Mysterious Cities of Gold and Dogtanian - I have to skip after about episode 2 or 3, and not watch again until the final episode, which gets given the full reverence. They're just too long, and in some cases, as I live in an apartment block without very good soundproofing, may lead to my neighbours either getting annoyed and/or thinking I'm a lunatic (they may be right, but I don't want them THINKING that). Just think about how much time you're going to end up watching that same repetitive sequence over the course of a 26 or 39 episode series (heck, the 13-ep Eden Of The East was bad enough, I skipped it about half the time). The 90-second ones aren't SO bad, you're only going to blow about 45 minutes all-in, but some of the examples given here are THREE OR FOUR MINUTES LONG. Good God! You shouldn't spend as much time over a whole season's run watching the intro/outro as would be taken up by a full episode (or two, or three). That's just insane.
(In some cases, the intro themes were even released as singles (Nadia of Blue Water, The Raccoons)... and the longer ones didn't even need to be extended). If it's being shown on TV once a week, that's not so bad... two or three a night when ploughing through a show on disc, it's just too much, no matter how good it is.
The Evangelion
outros get a free pass though, simply because of how often they changed the music...