Animals of Farthing Wood - Fantastic cartoon, but wouldn't get aired now for love nor money. Too sad for the little kiddys because heaven forfend they realise cute little animals die.
Grave of the Fireflies - I can't remember where I first heard about it, but I bought it knowing it was going to be a downer, but I honestly didn't expect it to be that much of a downer. I want to recommend it and lend it to my friends, but I'm not that cruel.
Full Metal Alchemist - The only time one of my friends has admitted to welling up is when watching this. Not going to say more, but if you've watched it, you know the places and if you haven't, watch it now.
The Snowman - I don't know why, but I always found this depressing.
When the Wind Blows - Watch this and you'll want to top yourself. Originally a comic, its about an old couple in the lead up to, and the aftermath of a nuclear strike on London. Its probably the only film I have thats too sad for me to watch all the way through.