I found the animatrix shorts Second Renissance to be that way. Like I said, I never finished Chobits. All I have is the second DVD. I do remember Chii freaking out in one of those episodes though and all the other Chobits went offline
I know Saikano was a very sad anime. I just got a dvd preview of the first episode and felt like I was going to cry. I never bought it but it looked like an interesting story.
Yeah, Saikano is horribly sad. But seriously if you want to cry watch: Elfen Lied, Shakugan no Shana, This Ugly Yet Beutiful World, even End of Evangelion.