Revealing mistake: Before Barney jumps out from behind the crate to surprise the two cops in the alley you can hear the director yell out "Don".
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)
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)
Three Wishes for Opie - S5-E14
Revealing mistake: Barney, convinced he can foretell the future, is in the darkened jailhouse, seated between Goober and Floyd at a table. Goober strikes a match and lights a lamp which is directly in front of Barney. The immediate illumination obviously comes from a studio light because the men's shadows on the wall are to the far left of the men, instead of directly behind them.
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: Barney and Mr Weaver scuffle in the dark. When Andy enters the store he turns on the light, but the light switch is still in the off position.
Revealing mistake: When Barney blows the dog whistle to call Blue from the back room, the dog comes running and knocks Barney on his back, and we can see a bit of the cushion that's under Barney's back - it's visible within the shadow under him.
Revealing mistake: When Barney and Andy are searching the woods for an escaped con, they stop to rest and you can see the light level of the woods go up and down as the crew tries to adjust the reflector screen for the desired lighting.
Mayberry Goes Hollywood - S1-E13
Revealing mistake: When Andy is listening to the Hollywood producer talking about how he wants the town back the way it was before, over Andy's shoulder the "window" in one of the stores is blowing in the wind.
Revealing mistake: On most of the corners of the buildings, you can see seam lines running down between the different sides of the buildings. The bricks on the opposite sides usually don't line up with each other either even though they are supposed to be the same brick.
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.
Revealing mistake: The woman in the foreground, as the siren is stuck, is carrying a paper grocery sack from "Hollywood Ranch Market" which was located on Vine St and Fountain Ave in Hollywood, Los Angeles. (00:09:35)
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.