I thought that The Power of One was quite depressing. Even the ending did not make up for the life of crap that the lead character PK had to live.
Some that haven’t been mentioned yet:
**
Amores Perros (Mexican)
In the Company of Men
In the Name of the Father
Kids
Naked (British)
Night On Earth (Final Chapter)
Romper Stomper (Australian)
This Boy’s Life
**
Anyone who thought American Me was a rollicking good time just has to be a heartless bastard.
As I am sure none of you people are heartless bastards, I must conclude that none of you have seen it. That, or else you’re repressing the memory, for which I can hardly blame you.
I certainly like old movies and would recommend All Mine to Give. I don’t know how to describe it without giving too much away, but it’s a sad Christmas type movie.
Let me second Welcome to the Dollhouse . All the more depressing for being so realistic.
You don’t have to be gay to be horrifically depressed by Bent, a drama adapted from the stage about three gay men during the Holocaust. Not even Mick Jagger in campy drag can add a smile.
I thought the movie What Dreams May Come was horrifically depressing. (When Max von Sydow is your comic relief there’s no point even leaving the cap off the Prozac bottle.)
I think the OP nailed the two most depressing ones possible.
Isn’t Where the Red Fern Grows where The dog or someboy critical dies in the end?
My MIL bought it for the kids ( under 6 years of age to watch) and I just remember reading the book and being very sad at the end, but cannot remember why.
The Talented Mr. Ripley always depresses the hell out of me.
Oh, without a doubt. I came in here to recommend this movie because I sobbed my head off all through it. I mean, Grave of the Fireflies is sad, but it saves it’s biggest punch for the last 10 or 20 minutes. But Monster’s Ball punishes you practically from the beginning to the end.
Once Were Warriors is powerful, brutal and shocking, and I seem to recall it leaving me feeling depressed for several days, but I’m not sure about uncontrolled sobbing. I’m thinking by the final blow I was just way too numb to cry.
Still, all three movies assault the senses and blugeon to death any happy thoughts you may have once had. Have fun!
Schindler’s List
Okay, what do I win?
For an encore:
The Pianist
Spartacus
Braveheart
Das Boot
Dangerous Liaisons
Lawrence of Arabia
Romeo and Juliet
Heh, I was just gonna mention Jude. 
Two more:
Voyager, starring Sam Shepard and Julie Delphy, and directed by Volker Schlondorff (The Tin Drum). Based on the novel Homo Faber by Max Frisch. With the hopeless, existential finality of its closing scene, this is one of the most depressing movies I’ve ever seen. (Not on DVD)
Au Revoir Les Enfants, directed by Louis Malle. Not too bad until the final moments which will have you bawling like a little baby if you’re any heart.
Addendum: Au Revoir Les Enfants is not on DVD either.
One more:
Before the Rain, starring Rade Serbedzija and the late Katrin Cartlidge, directed by Milcho Manchevski. Religious strife in Macedonia = very depressing. Interesting note: the storyline is structured like the contemporaneous Pulp Fiction. (Alas, this also is not on DVD.)
I also recommend Once Were Warriors; got it on DVD and have trouble getting through it (despite how great it is).
Two I haven’t seen listed yet:
Margaret’s Museum: depressing life of a young woman in a 1940’s coal town. Even when you see the tragedy coming, it hurts. Might be harder to find- Canadian film.
Brazil: Terry Gilliam’s most dark and depressing comedy. Great visuals & bizarre concepts.
On the Beach
If you have beloved pets, then Gates of Heaven … that makes some pet lovers sob.
(Gates of Heaven, not to be confused with Heaven’s Gate)
Might not be common enough, though; at a Blockbuster, it might be filed in the documentary or artsy section.
Oh yeah, I just thought of “In the Bedroom”.
Brutal.