Generally speaking, I'll temporarily stop watching many animated series, for a number of different reasons, but this mainly happens when I don't have enough time, resources or willpower to watch everything.
However...most of the time, if I'm honestly interested in what I've been watching, and this follows no specific rule or logic beyond my preference for shows with an actual narrative as opposed to episodic ones...I'll make an effort to go back or even do a full re-watch later. Once a show has got me completely hooked, sooner or later, I think it earns the right to at least that much.
Still, to name some specific examples of series I've voluntarily and permanently stopped caring about:
The Simpsons: I would say that this show will always represent a lot of fond memories for me, thanks to growing up watching it, but I've realized how repetitive it is and just can't possibly keep up any longer. I might catch bits of a random episode, rarely, and actually did go watch the movie, but that's it.
South Park: I was never a huge fan, but for a while I did make an effort to get into it. What seemed to be occasional sparks of brilliance grew old and the rest of its humor was too juvenile in the first place.
InuYasha: I had problems with my cable provider during my original attempt to watch this. That frustration made me want to see the series even more, whenever the opportunity arose. I really wanted to like it, considering the cast was interesting enough and the basic quest looked like it was getting somewhere at times, but then the damned thing dragged on way too much and, as I soon learned through spoilers, apparently wasn't even worth it. The decision to not watch anymore came easily.
Ranma 1/2: See InuYasha. Replace "basic quest" with "romance" and it's the same thing all over again.
Shounen Jump series: This isn't too specific, but after being a huge Dragon Ball Z and Saint Seiya fanboy, I've probably seen enough ridiculously long fighting shows for a lifetime. They're too repetitive, taking way too long to get anywhere and their endings usually go from redundant to pointless, especially whenever a new series or season is introduced. As a result, I have no real interest in Bleach, Naruto or the genre as a whole. They might actually end properly, but their basic formula doesn't seem different enough for me to try and wait for years.