Factual error: After the interview between J. Edgar and Tolson, they go to a well known D.C. area department store. The exterior of the store shows, "Julius Garfinkle & Co." but on the credit application it reads, "Julius Garfinckel & Co." It was founded as, "Julius Garfinkel & Co." in 1905 even though the owner's name did indeed have a "C" in it.
Continuity mistake: J. Edgar takes his secretary-to-be on a date and tries to impress her by showing off his card system for cataloguing books in the Library of Congress. When he runs to the book shelves he brushes past an open book set up on a display stand - sitting on a sheet of glass. The glass moves when he touches it and hangs off the edge of the table. When he returns seconds later the glass is back in place, centered on the table. His date did not move; she was still standing where he left her a few feet away and they were the only two people in the Library.