It was said by tsq and several others, but Grave of the Fireflies was also the first movie I thought of upon seeing the title. Holy jebus, that movie is a metric fuckton of depression.
Life Is Beautiful. Something about the humor Benigni brings to it makes it even harder to swallow than Schindler's List Braveheart probably deserves a mention in here, And Million Dollar Baby, which I can't even watch, because it's just too hard
Ironweed is near the top of my list, already mentioned upthread. One of the things that makes it so great is that it is a personal tragedy that doesn't essentially require any grand events for its backdrop. It could happen to anyone, anywhere, anytime. Nicholson and Streep are, as reputed, outstanding. But it's a real downer.
I love watching sad movies. Anyone catch The Impossible with Naomi Watts and Ewan McGregor? Wow, depressing and sad. I bawled my eyes out during a scene where he makes a phone call home. Very sad, but I highly recommend the movie. I don't know if it's still available on Netflix but I'll never forget this documentary, Dear Zachary: A Letter to a Son about his Father. Holy moly. Sad, angry, depressed, it makes you feel the whole gamut of emotions. True story too.
Les Miserables (2012). I mean, Hathaway had only 15 minutes of screen-time and she turned me into mush.
My two votes would be The Virgin Spring: http://en.wikipedia.org/wiki/The_Virgin_Spring and Bless the Beasts and Children: http://en.wikipedia.org/wiki/Bless_the_Beasts_and_Children_%28film%29 Also disturbing was what I consider to be Boris Karloff's best movie--Targets http://en.wikipedia.org/wiki/Targets In the film's finale at a drive-in theater, Orlok — the old-fashioned, traditional screen monster who always obeyed the rules — confronts the new, realistic, nihilistic late-1960s "monster" in the shape of a clean-cut, unassuming multiple murderer. Too real.
A Soldier's Girl - A Showtime TV movie, but a true story. It is as equally shocking as it is sad. I was crying for a long while after it, and it's certainly not a film I think I could watch again. Once is definitely enough. It's also sadder to me because it's quite relevant to my life.
I actually cried at the end of the original Godzilla when they sucked the oxygen out of Tokyo bay and killed him.
Another for Grave of the Fireflies. It hits me a bit hard since my Dad flew fighter-bomber raids on Japan in 1945.
Shadowlands with Anthony Hopkins as C.S. Lewis. The man finally finds love only to have it ripped from him by cancer, and the personal conflict he endures as he comes to terms with the pain of loss. It's gut-wrenching.