I'm surprised no one has mentioned The Last Unicorn!
"Why do you come to me now?" had me tearing up, and the ending on the beach, with Prince Lear awakening, saw tears rolling down my cheeks. Even just listening to the opening theme makes me tear up, actually. This, very recently, despite seeing it about 50 times as a kid.
Also, despite efforts to maintain control, I cried 3 times during UP. Could have cried even more, if I'd let myself!
As a teen I cried during the Anastasia cartoon, which is pretty ridiculous, but... I was in a sensitive phase. Yes, that's it.
How could I have forgotten The Last Unicorn?! I adore that movie, and it makes me cry too. When Molly yells at her, when she sees Lear fallen and turns on the bull, when Lear wakes on the beach, and even when she's talking to Shmendrick at the end ("I am a little afraid to go home. I have been mortal, and some part of me is mortal yet. I'm no longer like the others, for no unicorn was ever born who could regret, but now I do. I regret...") ::sniffle::
I'm curious as to what was the third thing in UP! that made you cry. Without using spoilers, I cried within the first five minutes (duh) and near the end when he sees the pictures in the book.
And I cried during Anastasia when she and her grandmother realized they found each other (: