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.