Factual error: When Columbo initially latches onto the radio dilemma (which ultimately solves the case) the dial is tuned all the way to the left. He later tells someone that the radio is set to 'Classical 52', but an AM radio starts at 54, meaning 540KHz. There is no 52.
Character mistake: Columbo talks to the cardiologist. The doctor gives him information about his patient, Mr. Williamson. Columbo has no warrant, so the doctor is violating privilege. He wouldn't give that information to Columbo.
Continuity mistake: When Columbo meets Goldie for the first time she is having a massage. She then proceeds to dress by tying a dressing gown with a bow at the front. In the next shot the dressing gown is open with the tie at the back.
Continuity mistake: When Columbo confronts Markham in the lecture hall and helps him erase the chalk board, most of the word "architecture" remains, visible over Columbo's shoulder. In the next shot, it disappears. (00:31:00)
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