What Makes a Character "Likeable"?

I recently posted about Icy Sparks and how irritated I was by her wandering mind and her tendency to make up stories. I noted that I thought this was probably a bit of a survival tactic, but how I still wanted to slap her upside the head and tell her to quit lying and pay attention. I know. Endearing, right?

Shortly after that post, I posted a comment over at Chain-Reading about Pi, the main character in Life of Pi, which I loved. The same quality that I found so ultimately irritating in Icy is what endeared me to Pi. So, this left me thinking, "Huh." I know, Deep, right?

Anyway, on a base level, they were both doing the same thing, inventing stories, changing their realities to make their current situations more tenable, so why did I have such severe and opposite reactions to them? Granted, Pi's situation was one of true survival-stuck on a life raft for 200+ days-while Icy's was just one of endurance-make it through another day of mean kids and mean teachers and wondering, "WHAT IS WRONG WITH ME??" but still...Icy should be the more relateable-we've all had to endure feeling this way to some extent, and maybe that's the problem. Everyone's felt rejected, like a freak, like a pariah at some point, and I think most people manage to get through it without turning into irritating liars. I, on the other hand, have never had to endure 200+ days stuck at sea in a life raft with or without a bunch of wild animals. There's no telling what I would have to do or, more importantly, be able to do to survive, so judging Pi seems a bit harsh, especially when you realize the vast difference between his potential reality and his story. I don't mean to be cryptic, but I don't want to ruin anything for anyone who has not read the book.

So. Can you think of other characters/books/settings/plots to which you have had similar opposing reactions?