Revealing mistake: While Barney's explaining to Andy his new interest in realty they both walk to the back room, and just as they pass through the doorway the camera tracks too far to the left, so we can see the edge of the set wall and the rest of the back wall is missing. (The following shot is from the direction of that missing wall).
Aunt Bee's Invisible Beau - S5-E27
Revealing mistake: When Clara and Aunt Bee are talking in the kitchen, in the closeups facing Aunt Bee we see the counter, the cabinet, and the wall, but what we don't see are the curtains, the window, or the windowsill, which all should be on that wall. The fake window wasn't placed on the set wall for Bee's closeup.
Revealing mistake: When the bus pulls up for the Taylor family to leave for Hollywood you can see behind the bus a California freeway with cars moving in both directions. I don't think that Mayberry had a freeway right outside town. The scene is fixed when they return in "Aunt Bee on TV." There has been a tree put there so the road wouldn't show.
The Taylors in Hollywood - S6-E8
Revealing mistake: While the Taylors are watching the scene being filmed of the attempted jailbreak Gavin MacLeod plays the sheriff, but while he's fighting the Calhoun brothers it's the actor's stunt double who's actually doing the fighting. (00:17:45)
The Taylors in Hollywood - S6-E8
Revealing mistake: When Andy, Aunt Bee, and Opie are visiting the studio set while the scene between fake Aunt Bee and fake Andy is being filmed in a fake Taylor living room, note that behind the real Taylors we can see bits of the interior of their Mayberry house scattered in the background offstage. Some things to look for include their living room's stone wall with the mantle, their two reddish chairs (one with the fringe at the bottom), and a wall with their distinctive gold/silver design wallpaper. It's particularly funny when Opie looks at the "fake" living room set and comments, "Is that our house? It sure looks different!" while we the viewers can see the "real" Taylor set behind him. (00:20:35)
The Taylors in Hollywood - S6-E8
Revealing mistake: While Andy, Aunt Bee, and Opie are at the Hollywood studio watching a scene being filmed, when Andy mistakenly stands up and interrupts the filming we can see the wall and shelves behind the director, and it should look familiar because it's the wall and shelves from Floyd's shop in Mayberry.
Revealing mistake: When Andy arrives at the studio for the publicity shots, while he's speaking to Mr. Michaels note that behind them we can see the Taylors' Mayberry kitchen cabinet/counter and a wall with their distinctive gold/silver design wallpaper. (00:03:50)
The Legend of Barney Fife - S6-E18
Revealing mistake: When Andy and Goober enter the Taylor house Andy calls for Aunt Bee, and just as she walks out of the kitchen we can see the end of the set wall inside the kitchen. (00:14:20)
Revealing mistake: While Goober is up in the tree with the fake owl, when Andy and the Darlings leave the house they are standing on the porch, and after seeing the 'second omen' the Darlings leave the porch and head to their pickup truck, but what's funny is that we can see their reflections in the window behind Andy as they all walk off camera, turn around and just stand there, and we can still see those actors' reflections in the shots facing Andy and Aunt Bee even as the Darlings supposedly drive off.
Revealing mistake: During the scenes with Opie and Dolly in Walt's barn, the horse is wearing a halter with a ring under the adjustable chinstrap, and in every closeup a thin wire (like picture wire) can be seen attached to the ring as it's being pulled from offscreen to have Dolly turn her head away from Opie. Then later when Walt returns and tries to feed Dolly, once again a thin wire attached to Dolly's halter is pulled from offscreen, so she will turn her head away from Walt. (00:05:55 - 00:18:20)
Aunt Bee's Big Moment - S8-E23
Revealing mistake: In the close-up shot of Aunt Bee in the airplane as she starts the roll for her solo flight, as she passes out of frame you can see the inside door latch hardware, revealing that the door has been left open or removed for the shot.
Revealing mistake: At about 1:15 into the show, as they walk past the train station ticket window, you can see daylight between the wall of the ticket window and the rest of the building, showing it's just a prop building. It's fixed the next time they walk past it.
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.