Plot hole: When Jeannie and Tony were going to get married Jeannie said she could not be photographed. But in earlier episodes she could.

Factual error: Major Healy is an astronaut - and an Army officer. In those days, astronauts had to be pilots with thousands of hours of experience in fighters and other high-performance aircraft and the Army did not (and does not) have any such aircraft, only helicopters and a few small propeller planes. Astronauts could be Air Force, Navy, or Marine Corps pilots, but not Army. In fact, according to the insignia on his uniform, Major Healy is not even in Army aviation, but the Corps of Engineers, which is responsible for bridges, roads, dams, etc.

Revealing: During the episode Tony and Roger are dropped in the desert to see how they can survive for a week. At one point Tony orders Jeannie to stop interfering. She pops out and Tony spots a "large cat" up on a rock. Close up of the cat, shot of Tony picks up rock, shot of cat leaping. You can see a rope tied to the cat.

Visible crew/equipment: At the end when Major Nelson is in the magic show, and Jeannie makes him float in the air, you can blatantly see the bar on the levitation device stretching from Major Nelson and disappearing into the curtain.