(at around 41 mins) J. Edgar Hoover stands to make a speech and is seen buttoning the top two buttons of his suit jacket. After cutting to show Secretary Mellon, the camera cuts back to Hoover, whose jacket is unbuttoned with one hand in his pocket. He then removes his hand from his pocket and buttons up the jacket again.
Hoover making the dinner speech warning about "organized crime" is a total anachronism. Hoover denied the existence of organized crime for decades until approximately the early 60s.