When Col Parker is setting at his desk in 1962, there are several Elvis albums on the shelf behind him. One album, Flaming Star, was not released until 1968, in connection with the Singer TV Christmas Special, when Elvis appeared on TV.
When Elvis is at Fort Hood, Texas, his buddy tells him that there are provisions to allow enlisted men to "live off base." The Army doesn't refer to their installations as bases, they are called posts, which is the term a solider would have actually used.
When Elvis and Priscilla are in the car on their way to the airbase. Elvis' rank on his uniform is Staff-Sergeant, E-6. He only achieved rank of E-5, Sergeant at the end of his enlistment. He was only enlisted for two years total.
In the scene where Elvis is at the airport in Germany preparing to return to the United States, the Military Policemen guarding the area are wearing obsolete brown uniforms of World War II and Korean War vintage. They should have been dressed in the later Army green type like Elvis was wearing.
When Elvis talks to Colonel Parker about making 'West Side Story', there is posters of 'Follow That Dream' and 'Blue Hawaii' on the wall. Actually, those films wasn't even shot in that time, and 'Follow That Dream' wasn't even in pre-production.
When Elvis and his friends are target shooting, the outward-shredded bullet holes in the paper targets indicate that they were shot at from behind. The holes should have been clean and round, with little or no tearing.