The movie is set in the early 1950s, but at 50 minutes and 19 seconds in, when Sonny is driving through town at night just after he first has sex with Mrs. Popper, a car from the mid-1960s or later is briefly visible.
When Sonny and Jacy are driving to Oklahoma to get married, the highway markings are incorrect. In 1952, Texas highways were marked with a solid line down the middle of the highway with no-passing zones marked with dashed lines.
At the very beginning of the film Sonny is having trouble starting the old rusty pickup truck. After getting it running he starts to drive off. The soundtrack has him changing gears but we still see him with both hands on the steering wheel.
Just before Jacy goes off with Lester to the pool party, she and Duane are making out in the car. She moves her legs across the seat of the car and is clearly wearing light colored, flat soled shoes with bows on the top. When she takes her shoes off at the pool party, she is wearing shoes with no bows.
When Duane returns to Anarene toward the end of the film, he is on leave from the US Army in the middle of the Korean War, around 1952. His hair is clearly too long to pass Army regulations, which limit the length of a soldier's hair, after he has completed basic training. His hair would be too long even for today's Army. Moreover, Duane has sideburns, which the Army would never have permitted.