Revealing mistake: After a shootout with Baby Face Nelson at a hotel, Melvin Purvis converses with J. Edgar Hoover over the telephone. As he says, "Our type cannot get the job done." Watch the background. You can see a modern era train move along the tracks in the darkness. This is inconsistent with the film's setting of 1933/1934.
Revealing mistake: In an aerial shot of Lincoln Ave in Chicago outside the Biograph Theater the rubber mats laid on the road to look like 1930's brick streets only cover the middle 1/3rd of the street. On the outside 2/3s the street is modern asphalt and shows striping tape for bike lanes.