A Deadly State of Mind - S4-E6
Plot hole: Collier damages a post with his car as he leaves the driveway. His car would certainly have been damaged. The storyline is not long enough for Collier to have his car repaired. Columbo knows about the damaged post, but he never looks into how the damage occurred and never looks at Collier's car, which would have been incriminating.
A Deadly State of Mind - S4-E6
Plot hole: Collier's relationship with Nadia Donner is wildly improper, and her husband knows about it. It seems unlikely that nothing was done about it.
Other mistake: In the cabin, Mrs. Danziger says he can pick up a pair of golf gloves in Acapulco. Golf gloves are normally sold as singles, not pairs. Very few people wear two golf gloves except in quite cold weather.
Character mistake: The crew of the ship is British. Most of them refer to Columbo as "Leftenant," but the doctor switches between "Leftenant" and "Lootenant."
Other mistake: The captain and the doctor play very fast and loose with medical records privacy, letting Columbo see them and discussing them.
Factual error: The nurse is smoking a cigarette in the sick bay. By 1975, when the show was recorded, that would not happen.
Factual error: The nurse brings Danziger a sedative and takes his blood pressure. She has her stethoscope in her ears backwards. A qualified nurse would not make that mistake. When she comes in after the murder to do it again, she has the stethoscope in correctly.
Audio problem: The clock striking wakes Columbo, and he gets out of bed. We see that he is barefoot. He takes a few steps in the barrack room, and we hear the sound he would make if he were wearing shoes. Then the sound gets louder and has an echo when we see him outdoors on the parade ground (in shoes).
Answer: In the show, he was already a Lieutenant. A promotion would put him in the next rank up, which for the L.A.P.D. would be Captain. However, some of a Captain's duties would be overseeing other officers and ensuring they're compliant with policies, regulations, and standards. It would also most likely take him out of the field. This is something Columbo has no desire for as he rarely goes to police HQ's. Nor does he show interest in compliance and standards (for example, not going to his semi-annual evaluation at the firing range). However, he could still be assigned to a higher pay grade based on expertise, which is a form of promotion that does not include rank advancement. This would be going from Lieutenant I to Lieutenant II. I don't believe in the show it's ever started what his pay grade is. Although, in s02e01 (I believe) he mentions making $11K a year. Whether or not this was a true statement on his part, if you could find pay scale information for an LAPD Lieutenant in the 70's, it could give you an idea of his pay grade.
Bishop73