Episode #35 "Ricky Asks for a Raise," Lucy, Fred and Ethel do the quick-change act to get Ricky's job back. One of the three is seated at a table supposedly for 12, but there are only 4 places set at the table! (thanks to email@example.com)
In the episode "Breaking the Lease" #18, Ethel plays such songs as Sweet Sue. In a later episode (#91 "Lucy's Club Dance"), Ethel can definitely not play the piano. In "Ragtime Band" #173, Ethel can't play the piano there as well, and then in the Lucy Desi Comedy Hour with Ernie and Edie Kovacs, Ethel plays the piano accompaniment for Edie's song.