From Garrett Levin:
This year, the following films are all nominated for four or more Oscars: Black Swan (5); The Fighter (7); Inception (8); The Kids Are All Right (4); The King’s Speech (12); 127 Hours (6); The Social Network (8); Toy Story 3 (5); True Grit (10); and Winter’s Bone (4). If any of those films happens to win every Oscar for which it has been nominated, it will join a very exclusive club. Name the five films that have won every single Oscar for which they were nominated (minimum of four nominations). And which is the only one of the five to not win Best Picture?
Pictured: Ben-Hur (1959), which earned twelve nominations. It came close to running the table, taking home eleven statues (losing in the category of Best Adapted Screenplay to Room At The Top).
"The Return of the King"
Yep, that's one of them. In fact, it is at the top of the list with 11 wins in 11 nominations. It won for: Art Direction, Costume Design, Directing, Editing, Makeup, Original Song, Original Score, Picture, Sound Mixing, Visual Effects, and Adapted Screenplay.
Gigi (1958), went nine for nine. At that time, it held the record for most Oscar wins. It was short-lived, as Ben-Hur exceeded that total the following year.
Yep Gigi is tied with a film from the 80s for second most wins by a film that is batting 1000 (wins for all nominations).
The Last Emperor
That's the one that's tied with Gigi, winning all 9 of the awards it was nominated for in 1987.
There's a bit of a drop-off to the last two. One of which went 5 for 5 (and is the only one on the list to win the Grand Slam: Picture, Director, Actor, Actress and Screenplay) and the last one went 4 for 4 (and is the only one on the list to not win Best Picture).
The only "grand slam" winners are It Happened One Night, One Flew Over the Cuckoo's Nest, and Silence of the Lambs. So one of the remaining answers has to come from that list.
It Happened One Night went 5 for 5 and The Matrix went 4 for 4.