When Carmine goes to New York, they show the Twin Towers, complete and with the antenna. The show is set in 1967 (from the opening credits New Year's Eve segment). Construction of the first tower did not begin until 1968, the second in 1969, and they were not completed until 1971.
The show is set in 1967 as Carmine is in New York City auditioning for "Hair". A street cutaway shot shows mid-70s model cars and a theater marquee for Linda Hopkins in "Me and Bessie", which premiered in Washington, DC, in 1974 and did not open in New York until 1975.