Add a vote for "Grave of the Fireflies".
Another animated film I'd consider is "The Plague Dogs". Not as good as Grave, IMO, but it is depressing as hell. Especially if you like dogs. The book (by Richard Adams - the Watership Down author) is worse though. Lets just put it this way - the main characters are two dogs used in animal testing. Then it gets worse.
For live action films, I'd recommend "Mother Night". Also based on a book, this one by Kurt Vonnegut. This one opens in an Israeli prison cell, where the main character is being tried for crimes against humanity for his part as a propaganda artist for the Nazis.