Ward v Bob. Well, neither was in one of those few that got double digit AAs or near, like LOTR:RotK, Titanic, Ben Hur, West Side Story, English Patient, Last Emperor, Gigi, Slumdog Millionaire, say. Now Bond was in GWTW, which won eight competitive AAs, so that's a good start. And Bobby was in Deer Hunter (won 5) and Godfather II (won 6), also a good start. It seems to me that Bond was in a fair amount of highly acclaimed films that got zero Oscars, like The Searchers, but much AFI love. Same for Bob. Really, it would come down to an accounting for all those great films that pulled in one or two statuettes. No idea.
Also worth noting Leo DiCaprio is in quite a few with big AA numbers. And there's Peter O'Toole, who would be in the running with The Last Emperor, Larry of Arabia, and Lion in Winter, for starters.
Actually I'm starting to lean towards Leo.