The Return of Barney Fife - S6-E17
Continuity mistake: At the reunion you see a sign that says class of '48. In season 3 They had a reunion and it said class of '45.
Continuity mistake: When Andy and Floyd stop by Wally's station, Flora's wearing a striped uniform shirt and there are a bunch of cars lined up waiting for service. Then later, when Goober gets back from vacation and shows up at the station, in the first shot of the station Flora's wearing the striped shirt and all the cars waiting for service are the same as the earlier shot, but in the closeups Flora's now wearing a solid color uniform shirt. (00:06:05 - 00:09:20)
The Gypsies - S6-E23
Continuity mistake: While Andy and the townsfolk are at the courthouse during the storm, when The Gypsies walk in the two men are wearing wet rain ponchos and the woman is wearing a striped serape, but then she's wearing a wet raincoat, and it switches yet again in following shots. (00:21:45)
The Legend of Barney Fife - S6-E18
Continuity mistake: At the start of the episode, while Barney's in Andy's kitchen he's wearing a shirt with large distinctive designs, but when he and Andy walk into the courthouse Barney's shirt has completely different designs (though it's the same type of shirt). (00:01:00 - 00:04:00)
Aunt Bee Learns to Drive - S6-E21
Continuity mistake: The car which Aunt Bee buys from Goober has a rearview mirror which repeatedly disappears and reappears between shots, throughout the episode.
Wyatt Earp Rides Again - S6-E20
Continuity mistake: After Opie tells Andy that he knows he'll win at high noon, it cuts to Warren in front of Floyd's shop (with its furniture conspicuously missing), but later when Andy walks over to Floyd's shop its facade has changed. Note differences such as window structure, and the name "Floyd's Barber Shop" painted on the window which wasn't in the earlier shot. (00:18:05 - 00:24:15)
The Return of Barney Fife - S6-E17
Continuity mistake: When Barney and Andy are at the Class Reunion dinner, there is a band playing, but the accordionist's fingers aren't in sync to the music.
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)
Opie's Job - S6-E1
Continuity mistake: We see the exterior of the grocery/meat store a few times during this episode, and there are differences in things such as the actual facade of the building, the view through the doorway, etc.
Opie's Job - S6-E1
Continuity mistake: When Opie and Billy are both heading to the grocery store to apply for the job, the church is behind the boys and they're standing in front of the 'new and used furniture' store - both of which are to the right of the courthouse (when facing the courthouse), and to the left of the courthouse are the barbershop and the grocery/meat store.When the boys run across the street, they're coming from the opposite direction.
The Bazaar - S6-E5
Continuity mistake: When Andy looks out the courthouse window and sees Goober's pickup truck filled with the ladies auxiliary group, Andy rushes out the front door and there's a wall right behind him where there shouldn't be, and although there's a bulletin board on the wall it still differs from the real one, plus the wall's air vent and wall trim are missing.
Continuity mistake: Aunt Bee won a garbage disposal in "Aunt Bee on TV" and it's one of the gifts she keeps, but she buys another one in a later episode, "Lost and Found," with insurance money.
Continuity mistake: During the scenes at Wally's station, the two prop gas pumps each have just one fuel dispensing hose/nozzle which switch between being between the two pumps, or with their hoses/nozzles on the opposite side of the gas pumps.
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.