Factual error: In the scene where Anna is standing with George beside his father's GTO, Elaine suddenly runs in and pries her away from George. Elaine and Anna get into the cab. As the cab pulls away, the side door is marked as an NYC cab with a taxi license number, etc., but the license plate in the back is a California plate.
Honoria Glossop Turns Up (or, Bridegroom Wanted!) - S4-E3
Factual error: Sir Roderick has moved his psychiatric clinic to New York because the BMA wouldn't approve of his methods. Although we don't get exact dates for the episodes, since prohibition was in effect in the last episode and Puttin' On The Ritz came out in 1929, we can presume it's the late 20s/early 30s. The AMA wouldn't recognize psychiatry as legitimate medicine until the 1940s. Sir Roderick wouldn't have had any better luck in America.
Reptiles in the Rose Garden - S1-E1
Factual error: After the White House is teleported to the Mushroom Kingdom, the President can be seen still using his phone, but it would not work as it would be disconnected from the landline. (00:03:44 - 00:07:19)