Aunt Bee's Big Moment - S8-E23
Visible crew/equipment: After Aunt Bee's solo flight, just as the flight instructor walks over and gives Bee the papers as proof that she soloed, the large T-mark (rubber/plastic) is visible on the ground between Bee and the instructor.
Aunt Bee's Big Moment - S8-E23
Visible crew/equipment: After Aunt Bee tells Opie that she will solo the next day, when she sits in the chair to practice her flight maneuvers, the actor's tape mark is visible on the floor beside her feet.
Aunt Bee's Big Moment - S8-E23
Continuity mistake: When Opie asks his father how to spell the word "Renaissance," Aunt Bee holds a plate of assorted cookies, but when she stands up and heads to the kitchen the type of cookies on her plate have changed.
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.
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.