I usually don't stop watching until I've been bored with the show for quite a while. I feel like I've invested that much time in them, I should stick around and wait for it to pay off. Occasionally, I'll stop watching when I can't suspend disbelief any longer (bad acting or bad writing is the primary cause there).

I stopped watching General Hospital when I realized pretty much *all* of the characters had come full circle and were back exactly where they had been a year previous. I stopped watching ER when Abby got pregnant (although I'd been thinking about it before then). I've seen so many horrible pregnancy/birth plots that I just plain can't take another one, and frankly, ER tends to be among the worst of them.

I guess I don't tend to watch shows for only one character, so changes like that don't affect me much, unless I see them as a symptom of a bigger problem (e.g., Andromeda; it wasn't just one character who changed, it was all of them).