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.
"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."
At the end of the movie 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.