Just before Dewey's wife leaves, he looks at the empty couch and says he doesn't need them. When she walks out the door, she leaves two kids on the couch.
Title cards at the end say Dewey Cox was born in 1936. Scenes set in 1953 make several references to 14-year-old Dewey, which would make his birth year 1939.
As Dewey is performing his first song in the nightclub he starts playing high notes on the wrong side of the piano.
When Dewey's first wife is holding two babies telling him that he doesn't spend enough time with his kids, one baby is clearly a doll.
During his college show for "little man", contemporary cars are visible in the background.
At the school talent show, set in 1953, the lead guitarist plays a 1964 Harmony Bobcat electric guitar.
When Darlene talks to Dewey about quitting the LSD in the studio, Sabian drum cymbals are clearly visible in the background. The scene is set in 1966, the company was founded in 1980.
When Nate and Dewey discuss what they want to be when they grow up, Nate says he's going to be an astronaut. Although the term was used in rare literature as far back as 1930, the scene takes place in 1946, before the term 'astronaut' became a common term in the late 1950's.
When L'Chaim visits Dewey in prison, he begins talking in Yiddish, explaining that the guards won't understand what they're saying, and suggests that Dewey go to rehab. Dewey responds in German.
During a rehearsal, Dewey complains that The Incredible Hulk (1977) is beating The Dewey Cox Show in the ratings. Both shows air on CBS, so they would not be in competition. Plus, his show aired Thursday nights, and Hulk aired Friday nights.