Continuity mistake: When Opie surprises Andy by telling him that he wants to take piano lessons with Clara, he heads to the piano to play what he already knows, and when Opie plays Chopsticks on the piano he's wearing a different shirt.
Continuity mistake: After Howard's appearance on the TV show, when he and Andy get back to Mayberry and talk to Opie on the porch, both Andy's tie and Howard's bow tie have striped designs that are positioned differently once they walk into the house, which could only happen if they were retied. (00:15:00)
Continuity mistake: When Harry hangs the sign on Floyd's shop it reads "For sale Harry Walker real estate," but in the closeup of the sign the word "by" suddenly appears and it reads "For sale by Harry Walker real estate", then it disappears. It looks as if it reappears in the wideshot when Howard stops to read the sign later. (00:01:15)
A Visit to Barney Fife - S7-E18
Continuity mistake: While Andy and Barney are sitting on the porch, Barney's clip-on striped bow tie is clipped on the opposite way (stripes flip) when Ma and her family get back from "shopping." (00:14:45)
The Statue - S7-E23
Continuity mistake: At the end, when Aunt Bee concludes the reading of the minutes and mentions the improvement fund, she's holding the blue folder up in her hands with its notepapers inside, but in the next shot one of the notepapers is suddenly flipped over the top of the folder. (00:23:45)
Aunt Bee's Restaurant - S7-E21
Continuity mistake: While Jack is trying to help Aunt Bee understand her fear, in a few of the closeups of Aunt Bee and Andy (seated on the couch) note the fruit bowl has moved and the candlesticks have vanished from the server, the large figurine on the left side of the hutch has vanished, and the flowers from the table have also disappeared. Then everything is back in their place. (00:21:10)
Continuity mistake: When Helen and Howard are talking about the lack of funds for the school play Andy comes up with a solution, and as Helen walks up the street there's a brick wall adjacent to the grocery store window which shouldn't be there (shot on the studio soundstage not the backlot). (00:07:05)
Continuity mistake: When Aunt Bee and Opie drive up to Wally's station to get gas, the car's vent windows are either open or closed depending on the angle of the shot. (00:06:05)
Continuity mistake: During the baseball game, when Opie slides into homeplate we can see the concrete mixer truck in the background beside the building, but when Helen shows Aunt Bee the photo she took of Opie the truck is gone. (00:09:40 - 00:23:25)
Answer: It was a joke. They were supposed to be dumb hillbillies.
Both. According to later interviews with both Andy and the Dillards band (the real name of the band that played the parts of the Darling children, of which Charlene (Maggie Peterson - Mancuso was also a member) the reason the Darling boys never had any lines was because they would've had to have been paid if they spoke. In return for appearing without lines or pay, Andy promised to get as much of their music onto the show as possible which, he did. This obviously gave them a lot of exposure.