Factual error: Towards the beginning of the film, the Cashes receive a notice about their rent. It's the 1950's and the address has a ZIP code. ZIP codes were not introduced until 1963.
Continuity mistake: In the room when DJay first plays his keyboard, the dirt pattern on the door behind him changes drastically when the scene changes.
Continuity mistake: After Elvis's first live performance, his parents are discussing how proud they are. They and the crowd are still moving in time to his song even though he's already stopped, indicating the scene was originally intended to take place during the song.
Factual error: In the scene where Brian gets fired from The Rolling Stones, Keith Richards sees a live album Jimi Hendrix and mentions it, Brian says Hendrix sent him one. But this scene must have taken place in 1969: Hendrix hadn't released any live albums earlier than 1970.