In the 1960s, Don Knotts was sitcom royalty. He took home five Emmy Awards for his work as Barney Fife on The Andy Griffith Show, setting a gold standard for comedy acting that was poignant as well as hilarious. But for all the acclaim Knotts received in that decade, it didn’t guarantee him work in the years to come. By the 1970s, Knotts was still working steadily on television, but guest-starring on Fantasy Island and The New Scooby-Doo Movies was a clear step down.

Sitcoms had changed, and nothing could have been further from the gentle humor of The Andy Griffith Show than the bawdy double entendres of Three’s Company. That comedy, built around the lie that Jack Tripper was gay and therefore a “safe” roommate for Chrissy and Janet, relied on sex jokes, misunderstandings and the tight T-shirts sported by Suzanne Somers. The show proved so popular...