When Lt. Parker backs his jeep into Capt. Binghamton's jeep while Binghamton is standing in the front passenger area, in the long shot Binghamton falls out and lands with his head pointing toward the two jeeps. When Parker and 'Tinker' Bell jump out to assist Binghamton, however, in the closer shot Binghamton suddenly has his feet toward the jeeps and his head pointing away.
When Gruber is playing badminton, the balloon is getting away from the girl. The next instance she is hitting it back to Gruber.
When Parker hits the polo ball, it strikes Binghmton from a different angle than where he hit it from.
The very title is an anachronism. While the sitcom takes place during WWII (1941-1945 for the U.S.), Igor Cassini coined the term "Jet Set" in the 1950s, to celebrate the rich and famous who were the only ones who could afford to travel by the newly-introduced passenger jets. The term for the upper classes used in the 1940s was "Cafe Society."
While dancing with a blonde nurse in front of the "yacht," McHale tells her to "be more like Zsa-Zsa." Zsa-Zsa Gabor was unknown in the United States during WWII. Her first movie and TV appearances were in the 1950s.