Platoon -- I barely remember that; it couldn't have made much of an impression! But that long opening sequence alone qualifies Full Metal Jacket as a downer. The movie wasn't even nominated for an Oscar.
I think the first real downer I saw -- that is, the first that truly got to me -- was The Bicycle Thief. It was nominated but lost. That one left me wiped out, bowed down by the hopelessness so graphically shown in the movie. That's pretty much been my standard for downers ever since. Another one I'd put in that category is Jude, a beautifully made movie which also happens to be depressing as hell. It was completely ignored by the Academy.