When Jan and Jonathan are talking in front of the interior design store about the car he is offering her, the same extras are seen multiple times. A woman with a blue coat and gray hat walks by four times, and a woman with a red coat walks by at least three times.
Brad's watch disappears as he holds his coffee cup.
When Jan is looking at Brad's apartment before she redecorates it, Brad knocks a red pillow off the couch as he jumps up when Jan is investigating the switches behind the couch. When she flips the second switch revealing the hide-a-bed, all of the red pillows are back on the couch.
When Jonathan (Tony Randall) gets hit in the diner, he gets hit on the right side of his jaw (the man swings with his left arm), yet in his office in the next scene he has a bruise on the left side of his jaw.
A party line phone would not ring if any phone on the line was off the hook. To call another phone on the same line, a special code was dialed, then the phone was hung up which would cause the originating phone to start ringing. When the phone stopped ringing, the caller would know that the other party had answered. This is not how Brad does it.
A party line combines the phone lines for customers that are physically near each other, such as adjacent apartments or houses next to each other. For Brad to carry Jan from one building to another, each having hundreds of apartments, and with both of their apartments on the same party line, this is impossible.
When confronting Brad in the nightclub, Jonathan mockingly calls Brad "Tex" instead of his correct alias, "Rex." Jonathan is using "Tex" as a generic nickname for a Texan; he is not trying to use Brad's alias.
When Jan and "Rex" are at the piano bar singing 'Roly Poly', the man behind the pianist is strumming a four-string electric bass guitar, but the sound is from a six-string electric guitar.
When Brad sees his redecorated apartment, the cat continues to meow even though its mouth is closed.
When Brad is carrying Jan out of her apartment, the board on which she is supported can be seen.
When at the bar singing 'Roly Poly' with Jan, "Rex" lets his real accent slip a few times.