Audio problem: Magnum slips into the car in 1936 with his client out of the club. If you pay attention to the audio, there is sometimes a droning noise as if someone were using a leaf blower nearby (it's actually because in editing they cut out all sounds below a certain level - but some of it spiked enough to stay into the audible range). (00:11:20)
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)
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)
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)
Revealing mistake: In present day, Magnum is flying with TC to the old house. Selleck and Roger Mosley are talking inside the vehicle about being there before, and TC's shirt is dark red. However in the wide shot of the landing, whoever is piloting the chopper is wearing a yellow shirt. (00:43:40)