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.
Visible crew/equipment: After the boys meet Flip Conroy and talk about the football team, when they all disperse and Opie starts walking away the T-mark is visible on the ground.
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)
Other mistake: Right before Aunt Bee wins the five dollar prize at Wally's station she talks about how she's never won anything, not even the church raffle, and right after she wins she excitedly waves the five dollar bill and says, "I've never won anything before in my life!" However, in season 6 episode "Aunt Bee on TV" the man from the IRS stated that Aunt Bee had won $4,850 worth of prizes on the game show, and even though she sold most of her prizes she kept two of them. So her statement makes no sense even for Aunt Bee.
Other mistake: Twice during the episode the actor who plays Floyd is replaced by his body double. The first time is when Floyd's driving to and away from Wally's station in the earlier scene, and the second is later, when Aunt Bee leaves Floyd's shop and we see Floyd stand up through the window.
Visible crew/equipment: When Aunt Bee drives her car up to the gas pump at Wally's station and then wins the five dollars, the reflections of crew and filming equipment are visible on the surface of her car. Also, note that the chrome trim around the windshield is covered with black tape, presumably to prevent reflection. Additionally, during all the scenes at Wally's station, the back end of one of the quonset huts from the "Gomer Pyle, USMC" set are visible, either at the left side of the screen or in the background.
Continuity mistake: When Opie and Arnold rush into the house and tell Andy and Aunt Bee about the party, in the shots facing the boys we can see the reflection in the mirror behind them, and there's a lamp atop the credenza, but in the shots facing Andy and Aunt Bee that lamp doesn't exist.
Visible crew/equipment: At the start, while Aunt Bee is ironing the bowling shirt, a chalk T-mark is visible on the floor under Opie's chair.
Visible crew/equipment: When everyone is at the airport before the flight some chalk T-marks are visible on the floor, such as the one between Andy and Helen when Andy takes the picture of Aunt Bee and her friends. (00:08:45)
Other mistake: During the scenes at Elmo's shop, where Aunt Bee's pictures are developed, it's obvious that this is actually Emmett's Fix-It Shop. We can see the window of the shop next door through Elmo's doorway, and recognize it as the store next door to Emmett's shop which has "Furniture, New and Used" on its front window.
Visible crew/equipment: After Andy hatches the plan to have Aunt Bee and her friends pick up their photos at the same time, in the following shot just as Andy walks into the shop, the shadow of the boom mic is moving beside the doorway, at the top left side of the screen.
Visible crew/equipment: After Aunt Bee's flight lands, when it cuts to the hotel lobby there's rug on the floor extending from the front desk to the stairs, but in the next shot as Aunt Bee and her friends walk into the lobby that rug is gone (this doesn't refer to the small rug behind them). Also note the tape mark on the floor where the bellhop stops the luggage trolley.
Andy's Trip to Raleigh - S8-E4
Visible crew/equipment: When Andy gets back from Raleigh and shuts the car door, just as the camera moves in closer we can see the reflections of two crew members (the one wearing a yellowish shirt looks as if he's pushing the camera dolly), on the window behind Andy.
Opie Steps Up in Class - S8-E5
Continuity mistake: When Aunt Bee picks up Opie at Billy's house, she parks the car in front and walks up to the front door. In the wideshot there's a space of approximately 3-4 feet between the blue window shutter and the door trim under the pediment, but in Aunt Bee's closeup there is no space whatsoever.
Opie Steps Up in Class - S8-E5
Visible crew/equipment: When Andy walks into the living room at the Hollander's house, the white tape mark is visible on the floor between Andy's and George's feet.
Opie Steps Up in Class - S8-E5
Visible crew/equipment: When Aunt Bee picks Opie up at Billy's house, while Opie tells her all about lunch we can see the reflection of the camera and crew member, on the car's vent window.
Opie Steps Up in Class - S8-E5
Visible crew/equipment: When Aunt Bee drops Opie off at Billy Hollander's house, just as Opie opens the car door the reflections of a crew member and filming equipment are visible on the door's surface.
Opie Steps Up in Class - S8-E5
Continuity mistake: When Aunt Bee drives up to the Hollander house with Opie, in the wideshot the car's rearview mirror is affixed to the windshield, but in the closeup the mirror has vanished.
Visible crew/equipment: After the shot of Milly holding hands with Howard (holding a picnic basket), when it cuts to Goober sitting in front of the courthouse the camera angle is low, so the wood framework and lighting equipment at the top of the courthouse set is visible, just as Andy walks over to Goober. (00:05:30)
Continuity mistake: At the end, when Aunt Bee serves dessert her spoon and napkin are on the table, but in the next closeup of just Andy and Opie her spoon and napkin have vanished, then when Bee's seated she removes her napkin and her own dessert and spoon should still be on the table, but in the closeups of Andy and Opie they're not.
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.