I'd also mention Awakenings, an incredibly moving film. The scene where De Niro dances with the girl, knowing he's about to slip back into his sleeping sickness at any moment, is heart breaking.
But one of the most powerful and moving films I think I've ever seen is Lorenzo's Oil. When the parents (Nolte and Sarandon) first discover their son's illness and we see their devastation it is truly heart wrenching to watch and
almost brings a tear to my eye. Not much moves me to tears, I'm afraid. Although I am a big softy, I'm not given to bouts of crying.