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).
Continuity mistake: When Columbo is talking with a cadet, Columbo says that he was in the army. Later, when the Colonel mentions 2200 hours, Columbo asks him what time that is. If Columbo had been in the army, he certainly would have known that 2200 hours is 10:00 PM.
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.
Factual error: In common with a number of characters in numerous episodes, Columbo hopelessly contaminates crucial evidence - at the end of the episode he handles the surgical gloves used by the killer with bare hands.
Factual error: Several times, the revolver used to murder the singer is referred to as a "Weatherby" or "British Weatherby." Weatherby is an American manufacturer of rifles and shotguns; they do not make handguns. The revolver used is a Webley, which is of British manufacture.
Other mistake: When Rosanna's body is brought to the ship's medical room, the doctor confirms that there are no bruises on the body, but she had been forcibly slapped in the face earlier that day by Mr Danzinger (Robert Vaughn). The slap was hard enough to knock her over therefore a bruise should be visible, even through make-up.
Plot hole: The cruise ship is sailing from LA to Mazatlan in Mexico. Looking at a map, the ship will hug the west coast on its journey therefore is never far away from land. The murder takes place on the first evening of the cruise only a few hours after setting sail by which time they won't be far from San Diego. Why didn't the captain order the ship into San Diego or another suitable harbour and let the police take over the investigation?
A Deadly State of Mind - S4-E6
Continuity mistake: When Columbo asks if he can borrow the psychiatrist's lighter, the man has his cigarette in his mouth and a newspaper in his left hand as he reaches into his pocket for the lighter. In the next shot, as he hands the lighter to Columbo, the cigarette has jumped into his left hand with the newspaper. (00:27:40)
A Deadly State of Mind - S4-E6
Other mistake: Right before Columbo tricks Doctor Collier into incriminating himself, he places Collier under arrest, even showing him the warrant. At no time does Columbo inform Collier of his Miranda Rights; meaning that Collier's self-incriminating statements would not be admissible in court. Although the series typically features Columbo cornering the culprit in a way that is unlikely to hold up in court; this particular example stands out as Columbo has torpedoed his own case. If he had not placed Collier under arrest and let him incriminate himself then the statements (in front of witnesses) would be admissible as evidence.
A Deadly State of Mind - S4-E6
Plot hole: (Spoiler) Mrs. Donner screams in fright just as she jumps to her death from her balcony. This seems unreasonable since she believes herself to be going for a swim (under post-hypnotic suggestion) The scream is heard by a later witness to help establish time of death. If she suddenly realized the deadliness of her act, why would she continue with it?
Continuity mistake: When Raymond brings the doctor his milk and pill, the tray with the milk and pill are already on the bedside table.
Factual error: Columbo gives another detective $5 to go to the gun range for him. Technically, that is a bribe and it would cost both Columbo and the other detective their badges.
Factual error: All LAPD officers must re-qualify with their weapons multiple times a year. If indeed Columbo could not "hit the target," as he says to another detective, he would never have made it through the police academy, much less have become a detective and have been promoted to sergeant and eventually to lieutenant.
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