Chuck Berry is depicted as angrily pointing out the resemblance between the Beach Boys' "Surfin' U.S.A." and his own "Sweet Little Sixteen" at the time of his arrest for violating the Mann Act. Berry was in fact arrested on this charge in December 1959, and ultimately (after two trials and a failed appeal) sent to prison from February 1962 to October 1963. "Surfin' U.S.A." was released in March 1963, when Berry was still in prison.
The 45 RPM record was not widely used during the time the film was set. Until the late 1950s, 78 RPM records were the state-of-the-art at home and on the radio. Studios were pressing mostly 78's when Chess Records started, but they don't appear anywhere in the film.
Several guitars in the studio are much later models than would have been available at the time. A Gibson ES-335, introduced in 1958, is clearly visible in one scene set in 1955. A Gibson ES-175 in the same scene (and some set earlier) has humbucking pickups, which were introduced in 1957.
A 45 rpm record insert known as a 'spider' is shown as a part of the Chess Records sign hanging in front of the studio. The 'spider' was introduced in the 1960's, years after Chess Records opened in 1950.
During the scene where Etta James is singing "I'd Rather Go Blind", she is seen with the microphone in her hand at the beginning of the second verse of the song. The camera then shows her from behind so Adrien Brody's character can be seen exiting the studio and the microphone can be seen clearly over her shoulder still on its stand. A second later the camera cuts to her face again and she is somehow still holding the microphone.
Etta James is portrayed as never having recorded before she signed with Chess, but an earlier insert shot showing a Chess record moving up the charts includes her hit "Wallflower," recorded for Modern Records in 1955, five years before she started recording for Chess.