Factual error: At the movies, Magnum is watching a movie with Bette Davis, the first one she played in, "The Bad Sister" and comments "Keep it up, old girl, you just might have a future as an actress." Funny, but the dream is set in 1936, not 1931 when that movie came out. Implausibility of finding it in theaters aside, by 1936 young Bette Davis had already won her first Oscar. (00:30:30)
Continuity mistake: Marisa during the movie gives Thomas an appointment. When she says "That's where I hid the papers." the woman in the background is leaning against her left hand. The chat continues in the next shot but the woman's position is different and gone is the hand. (00:30:20)
Almost Home - S3-E10
Other mistake: When Magnum passes a can of nondescript soda to Bridget outside of the Navy building after the chat with Maggie, the vending machine has a big triangle of masking tape to conceal the Coca Cola logo for sponsorship reasons - however the production failed to cover the Sprite button which remains recognizable. (00:13:00)
Other mistake: The clock in the millionaire's limo has all hands (including the second hand) perfectly still. (00:02:00)
Deliberate mistake: During the episode, whenever daily newspapers are shown, they lack any indication of the date.
Did You See the Sun Rise? - S3-E1
Continuity mistake: Magnum's sling looks different before the flashback and after it when he speaks with Higgins. (00:48:50)
Factual error: Magnum gets to the airfield looking for TC the first time, at the very beginning of his drive you can see in the building in the background a modern window AC unit - that kind of device was available in 1936, but its cost certainly makes it unlikely, as it'd cost the equivalent of $8,000 today, not really something for the dingy shack it's mounted on. (00:16:55)