Emma Goldman is interrogated in a scene early in the film, set in lower Manhattan in 1919 or 1920. Through a window, we see the Statue of Liberty to the right, and the Verrazano Narrows bridge to the left. This bridge was not built until the mid 1960s.
During telephone conversations, the corded handsets had a modular jack. These modular jacks did not become popular until the late 70's through the 80's. Until then the hand sets were hard wired directly to the telephone base.
Hoover and Tolson are in Del Mar, and gossip about having seen Desi Arnaz and Lucille Ball together at the racetrack before the two had even met. Lucy and Desi did indeed have a home in Del Mar, but not for 20 years after the time of this scene. Furthermore, they gossip about Lucy's fake red hair - she was not a redhead until 1943, having brown or dark blonde hair during most of the 1930s. Additionally, Hoover refers to her hair as "ginger"--a British slang term unknown in the U.S. in the 1930s.
In one scene set in the 1930s the Federal income tax department is referred to as the IRS. The name "Internal Revenue Service" was only given to that department in the 1950s. Previously it was called the Bureau of Internal Revenue.
As the 1933 Inauguration parade passes by Hoover's office in the background, outside his window, is the National Archives building. Its cornerstone had been laid, at most, one month earlier in February 1933 and the building was not completed until late 1935.
When Edgar's mother teaches him to dance, she has her right hand on top of his left shoulder and he has his left hand in the small of her back, which is the mirror image of the standard dance hand positions. If she had been leading, her right hand would be in the small of his back, not on top of his shoulder.
Near the end of the movie, when J. Edgar and Clyde are conversing with one another. J. Edgar is wearing a pair of suspenders with brass adjustment clips on his chest. These brass clips move from being parallel to being out of alignment and then back. This happens several times during the scene.
Hoover states that the Lindbergh baby's body was found in a spot that was within sight of the Lindbergh home. The body was found in Mount Rose, as shown in the film, but this is located miles from the Lindbergh home, which was in the Sourlands.
An early Tolson/Hoover scene at the tailors, is set up with an exterior shot of a well-known Washington department store. The stone engraving on the building says the "Julius Garfinkle" Company. The correct spelling of that Washington institution was "Garfinckel." When a document from the store appears moments later "Garfinckel" is spelled correctly.
The movie portrays the Lindbergh kidnapping as being the reason for the origin of the crime laboratory. In fact, according to the FBI website, the lab was much earlier. In addition, it was the New Jersey police and Schwarzkopf who did the work on the ladder, including the forensic work that led to the mill where the kidnapper was employed. Schwarzkopf, incidentally, is the father of General Norman Schwarzkopf, the commander of Desert Storm.
Neither Hoover nor Agent Melvin Purvis killed John Dillinger. Dillinger was actually gunned down by agents Clarence Hurt, Charles Winstead, and Herman Hollis. Most historical accounts give Winstead credit for delivering the fatal shot to the back of Dillinger's head. Ironically, given the film's depiction of Hoover as constantly claiming credit for the deed, Winstead received a personal letter of commendation from Hoover for it.
The horse racing scenes in the movie were filmed at Del Mar Racetrack supposedly in the 1930s and later in the 1960s. When the horses turn home you can see the track surface is not dirt, but a synthetic Polytrack surface. This surface was installed at Del Mar for the 2007 racing season. Del Mar racing prior to 2007 was run on a dirt track.
District of Columbia flags are displayed in congressional hearing rooms and in the Department of Justice building on various occasions. As these are federal buildings, the red and white DC flag would never be displayed in that manner. They are not District of Columbia offices, but federal offices. Moreover, the District of Columbia flag did not exist until 1938.