When Darlene is talking to Dewey about quitting the LSD in the studio, Sabian drum cymbals are clearly seen in the background. This scene was supposed to take place in 1966; the company was not founded until 1980.
At the end it says Dewey Cox was born in 1936. However in 1953 Deweys mother calls him her favorite fourteen year old son and it is mentioned several more times that he is fourteen. That would have made his birth-year 1939.
"The Dewey Cox Show" airs on CBS. During a rehearsal, Dewey complains that his show is getting beaten in the ratings by The Incredible Hulk, which aired on CBS, so it could not be in competition with "The Dewey Cox Show."
In addition to Dewey Cox mentioning his show was being beaten in the ratings by "The Incredible Hulk" - despite the fact both shows were on the same network - Dewey also mentions his show was on Thursday nights. "The Incredible Hulk" aired on Friday nights.