A Beautiful Mind
12 Monkeys

I'm not up on whether insanity qualifies as a handicap or not, but if it does then these two qualify.

The main character in the film and stage versions of The Light in the Piazza has a mental handicap. I haven't seen the film, but the adaptation that's currently playing on Broadway is one of the most beautiful love stories I have ever seen in my life. The movie stars Olivia de Havilland and (according to IMDB) seems worth checking out...of course, it's not on DVD. :-/