Continuity mistake: When Andy locks himself in the jail cell, the key is lying flat on the short wall, but after Barney locks himself in the other cell, the key has turned itself upright on its side.
Other mistake: After Barney gets annoyed over Andy's lack of confidence in his ability to be a sheriff, Barney gets on the phone with Sarah and asks to be connected to the mayor of Greendale, then it cuts to an exterior shot and the parked car has plate number "DC 269," the same as other vehicles in some earlier episodes.
The Farmer Takes a Wife - S2-E13
Other mistake: At the end, when Joe Waters parks his car in the sheriff's parking spot his car's license plate number is "DC 269," which has been seen in earlier episodes on other vehicles.
The Keeper of the Flame - S2-E14
Continuity mistake: When Andy holds Barney by the shoulders while congratulating him for an outstanding job of detective work, Andy holds his notebook in his left hand, but when it cuts to a different angle the notebook's suddenly in Andy's right hand.
Continuity mistake: When Andy and Barney catch up with the man who ran Fletch's truck off the road, the rearview mirror suddenly vanishes in the closeups that face the windshield on Bailey's car.
Visible crew/equipment: When Barney's reading the Sunday comic strip Andy comes in to close the jail in the courthouse, and when Otis walks out of his cell we can see the actor's chalk mark on the floor where he stands.
Character mistake: When Barney walks by the barbershop window he glanced in and waves at Andy who is getting a manicure. But the man in the chair isn't Andy, it's a stand in. (00:17:00)
Visible crew/equipment: When the new manicurist tells Floyd that she works only on commission, she lifts the small table and moves it to the window, and we can see the actor's tape mark where she had just been standing.
Other mistake: When Barney and Andy find the trailer that Malloy's hiding in with his partner and Horton (who's tied up), the other trailer that Andy and Barney are standing behind has license plate "DC 269," the same as some other vehicles in previous episodes.
The Merchant of Mayberry - S2-E22
Plot hole: In the episode where Barney confronts the two farmers for selling produce on the side of the road, Barney mentions that they're not allowed to sell because of "section 17.4" of the ordinance code - that street vending is not allowed, and Andy confirms it. But in this episode Bert comes to town and is selling dry goods, etc. and he's allowed to sell. Even Ben states that there must be a structure if he's selling - wouldn't you think that Andy and Barney would have known that already before setting him up to sell?
Visible crew/equipment: When Otis tries to escape from Aunt Bee's clutches by hiding in the laundry service van, just as Aunt Bee has the driver open the van's back doors we can see the reflection of the moving boom mic in the left window.
Continuity mistake: When Otis begins his 24 hour sentence at Andy's house by staying in the guest room, Otis is wearing suspenders (braces) that are sewn together in the back in an X-shape, but when Otis leaves the guest room he's wearing entirely different style suspenders attached to his trousers.
Other mistake: The new game/hobby boxes that Barney hands out to the Gordon brothers have their brand names partially covered. Mr. Potato Head has 'Head' covered, the brand name of the leathercraft set is covered, and part of the name on the metalcraft tapping set is covered.
Andy and Barney in the Big City - S2-E25
Factual error: Though they are suppose to be in Raleigh, the skyline view is not Raleigh, NC.
Visible crew/equipment: When Barney and Thelma Lou leave Mary's house the curved chalk actor's mark is visible on the floor by the door. Then Barney shows up again with his drum, and we can see a chalk T-mark on the floor. That same T-mark is also there in a later scene, when Barney barges in with the letter he found behind the file cabinet.
Continuity mistake: When Barney disguises himself as an old woman, in the very first shot he's wearing a dress with lace and tiny flowers, but in all the following shots it's an entirely different dress with large groups of flowers. (00:17:40)
Visible crew/equipment: After Andy and Barney walk out of Floyd's shop, when Andy makes a crack about Barney's ears being uneven Andy walks offscreen, then Barney shouts at him as Andy's presumably already down the block, but in their reflection in the window behind them we can see that as soon as Andy walks offscreen in this closeup, he actually stops and leans over where we see the camera's located.
Visible crew/equipment: When Barney and Andy are waiting for Virgil to arrive, just as the bus pulls up and stops, the reflection of the boom pole/mic is visible on the windshield. (00:03:50)
Revealing mistake: When Barney pulls the jail cell window out of the exterior brick wall, as it's being yanked out we can see that the brick wall is only thin drywall during the stunt, as well as the fact that the wall around the breakage differs significantly between the closeup and wide shots.
Other mistake: Barney's cousin is coming to Mayberry but gets off the bus in a previous town and misses it. Barney and Andy go out to find him, leaving from Mayberry. They are driving down the road when they come upon his cousin. The cousin is walking in the same direction as Andy and Barney are going, which would have meant he was walking away from Mayberry when he should have been walking towards 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.