Since I'm a big fan of Elfen Lied I would vote for it. There's many sad moments in there. Somehow I couldn't find any complete videos though, most are mixed up scenes with a song in the background <.<
Scenes I can think of would be:
Lucy as a child experiencing only cruelty
Mayu's story
Nana almost dying
and of course Lucy's "death" in the end
Was long ago when I last watched it but I gots dem DvDs :D It's a pretty good show, very gory and lots of naked moments.