Continuity mistake: When Aunt Bee and Opie drive up to Wally's station to get gas, the car's vent windows are either open or closed depending on the angle of the shot. (00:06:05)
Visible crew/equipment: When all the Darlings walk out of Andy's kitchen with a bag full of Aunt Bee's muffins, the actor's tape mark is visible on the floor, where one of the Darling boys looks down and stands. (00:20:25)
Continuity mistake: During the baseball game, when Opie slides into homeplate we can see the concrete mixer truck in the background beside the building, but when Helen shows Aunt Bee the photo she took of Opie the truck is gone. (00:09:40 - 00:23:25)
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)
Continuity mistake: When Goober and Warren are at the diner, things on the counter keep switching places while Goober eats his peanut butter, tuna, and ketchup sandwich. (00:09:00)
The Bazaar - S6-E5
Continuity mistake: After Goober complains to Andy about Warren it cuts to Warren walking down the street, and when he moves the bucket of corn in front of the grocery store, the window display and the taped signs change/vanish between shots. (00:02:05)
Malcolm at the Crossroads - S6-E3
Continuity mistake: At the start, Andy walks down the street and stops at the church steps to watch Ernest T. being a crossing guard, but when it cuts to the shot of Ernest T. we see the church steps yet Andy's gone, then when it cuts back to Andy he's still standing by the church steps.
Continuity mistake: After Clara cancels lunch with Aunt Bee, when Bee walks over to the table there are two candle holders on the server and a large picture surrounded by four small pictures hanging over the server. However, when Opie runs to the table, the left candle holder and bottom left small picture are both gone in his closeups. (00:19:40)
Malcolm at the Crossroads - S6-E3
Continuity mistake: When Andy rushes up to Ernest T. and tells him to quit throwing rocks at the car, Ernest T. holds the crossing guard sign with his left hand and points at it with his right hand, but in the closeup it's vice versa. (00:01:00)
Opie's Job - S6-E1
Continuity mistake: When Aunt Bee walks by Floyd's shop the barber's pole is missing, but when Opie and Billy run across the street there's a red/white striped barber's pole, then when Andy walks by Floyd's shop the barber's pole has red/white/blue stripes.
Continuity mistake: When Warren parks the squad car at the bus stop, as he gets out of the car note the long whip antenna is attached to the bracket over the rear passenger door, but when it cuts to the closeup the whip antenna has vanished. (00:21:25)
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.
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 Andy gets the phone call about Sam's victory, when Sam is pushed to give a speech, a few white tape marks are visible on the floor near the fireplace.
Visible crew/equipment: After Emmett talks to to the diner owner about having the bus stop moved to double his diner business, when Emmett walks down the block, the reflection of the moving camera dolly is visible in the store window.
Visible crew/equipment: When Andy, Howard, and Goober walk into Emmett's shop to tell him that Sam's running for head of City Council, the shadow of the boom mic is moving on the wall behind them, at the top of the screen.
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.