The ad agency's viewing room has both color and black-and-white TVs side by side. This was common at ad agencies in the 1960s to confirm that color commercials would also be acceptable on black-and-white sets.
The music used during the scene that Mrs. Fraleigh is having her baby in the limousine is the same music used in the pilot for 'The Munsters'. The pilot made in 1963 did not air for the public until 2007.
Dr. Boyer (James Garner) says that he helped Mrs. Fraleigh (Arlene Francis) to get pregnant after twenty years of trying unsuccessfully. This would have been just about impossible, because Arlene Francis, who portrayed Mrs. Fraleigh, was approximately 56 years old (and looked it) at the time the film was made, at least sixteen years past prime child-bearing age.
When older Mr. Fraleigh tells his underling to put a swimming pool in Beverly Boyer's (Doris Day) backyard before the commercial for the following night, the workmen are on site the next morning, constructing the pool, and it is complete that night. There is at least one problem which would make that impossible, curing (hardening) the concrete would take much longer than even 24 hours, which is double the amount of time depicted.
When Mr. Boyer is giving directions to Mike in front of the Cartier Hotel to the Doctor's Hospital to meet up with Mrs. Fraleigh to deliver her baby, he says to go up 3rd Avenue to 88th and east to the river. That is exactly how you get to Gracie Mansion, official residence of the Mayor of NYC.