The Taylors in Hollywood - S6-E8
Visible crew/equipment: After the studio's security guard tells Andy they're filming on stage 40, when it cuts to the Taylors walking onto the set we can see two chalk T-marks on the floor (one in front of Andy's feet). (00:13:00)
The Cannon - S6-E11
Visible crew/equipment: After Warren accidentally fires The Cannon, in the following closeup of Andy commenting that he shot a station wagon, the shadow of the moving boom and mic is visible at the lower left corner of the screen. (00:22:05)
Visible crew/equipment: After Andy agrees with Helen to suggest a blind-date for Warren, when it cuts to Warren walking into the courthouse, we can see set lighting and equipment behind the door instead of the usual painted backdrop. (00:01:55)
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.
Ernest T. Bass Joins the Army - S4-E3
Visible crew/equipment: When the two men are carrying the window pane past Barney and Andy, we can see the reflection of a few crew members' legs.
Visible crew/equipment: After Andy and Barney have asked Otis if he's seen the billy goat, when it cuts to Opie on his bike we can see the two white tape marks on the ground where Andy and Barney stand.
Opie's Job - S6-E1
Visible crew/equipment: When Goober walks up to Andy outside the grocery/meat store, before Goober picks up the apples we can see a chalk mark on the ground between his feet.
Visible crew/equipment: When Floyd and the other men are with John Canfield joking about Floyd's barber skills, a chalk T-mark is visible on the ground by the man with his foot on the bench. Then when Canfield walks away we can see more chalk marks on the ground.
The Bazaar - S6-E5
Visible crew/equipment: After Andy shouts at Goober to get his pickup truck away from the courthouse, it cuts to Andy, Warren, and all the women in the courthouse, and we can see the chalk T-mark on the floor at Warren's feet.
Andy's Rich Girlfriend - S3-E2
Visible crew/equipment: After Andy explains to Barney why he's taking Peggy to a nice restaurant 55 miles away, when Barney walks away from the cell we can see the leg and shadow of the tall wood ladder (presumably the crew's) at the right side of the screen, but when Barney and Andy are by the front door there's no such ladder behind Barney in his closeups.
Visible crew/equipment: While the fake band members begin to practice before the mayor shows up, in Andy's closeup right before he tries to stop Phil's flamboyant playing, the reflection of filming equipment is visible on the surface of Andy's tuba.
Visible crew/equipment: At the start, when Andy and the rest of the band are at Floyd's, we can see the black tape mark on the floor behind the barber's chair where Floyd's standing. Watch Floyd walk away from the chair for a moment, and when he goes back he looks down twice to make sure he's standing at his mark.
Visible crew/equipment: When Wilbur's in his car he thanks Andy and the rest of the men for all the shoe sales. As he drives off, the reflection of filming equipment is visible on the surface of the car, and when the car is offscreen multiple chalk marks are visible on the sidewalk, at the feet of the actors.
Visible crew/equipment: When the men have the misconception that Wilbur is a TV show talent scout, while they're at the barber shop talking about it, just as Floyd moves his legs to get up from the barber chair we can see the white T-mark at Jason's feet.
Mayberry Goes Hollywood - S1-E13
Visible crew/equipment: Before the producer's arrival in his car, when Barney's telling Andy that it'll serve him right if they don't put him in the picture because Andy's not wearing his new uniform, we can see the moving shadow of the boom's hinge and mic on Andy's shirt.
Visible crew/equipment: At the end, when Andy opens the door to Floyd's so Ed Sawyer can get a haircut, the set's large spotlight is reflected on the door's window.
Visible crew/equipment: While Floyd's napping in the barber chair, Andy's reading the newspaper, and when Barney walks in the actor's two curved chalk marks are visible on the floor, where he will stand.
Visible crew/equipment: While Barney's in the barber chair, Floyd starts talking about how he'd tried living at the back of the barber shop, and when the shot faces Barney we can see the reflection of a crew member's legs moving around - the only ones in the shop are Barney, Floyd, and the two men seated by the corner.
Visible crew/equipment: While Andy's in Opie's bedroom when Opie's thinking about running away, the moving shadow of the overhead boom mic is visible, first on the wall behind Opie's headboard and then on the door. Also note, when Andy shows Aunt Bee how to hold the baseball bat and moves her into position, we can see a long white tape mark on the grass (since it's possible that Andy placed the tape, however unlikely, it's being noted here).
Visible crew/equipment: After Aunt Bee plants the seeds of suspicion in Andy's mind about the shoe salesman, when Andy and Barney cross the street the reflection of the moving boom mic is visible on their squad's windshield, just as Andy walks off to the hotel.
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.