Deliberate mistake: After Barney buys the car from Mrs. Lesh, he and Andy are standing at the steps as he tells her not to look back at the car, and the next shot of Andy and Barney at the front door is the same shot from a few moments prior, but it's been flipped so they would be looking in the opposite direction. (00:09:30)
Character mistake: When Andy and Barney are looking through the yearbook, Barny's middle name is Milton. But Barney says many times throughout the series that his name is Bernard P. Fife.
Continuity mistake: After Ellie tells the Taylors that she's withdrawing as candidate, when she and Andy head over to the rally at the hotel they pass the church, and in the long shot a large sign has suddenly appeared at the front of the church.
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.
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.
Visible crew/equipment: When 'FBI Agent' Jenkins congratulates Andy and Barney on behalf of J. Edgar Hoover they're all standing near the door, but when it cuts to Jenkins turning around and they all walk over to the desk, we can see the two curved chalk lines on the floor where Jenkins' feet were positioned, as well as the curved chalk marks on the upper step where the mayor was standing.
Andy and the Woman Speeder - S2-E3
Continuity mistake: At the end, when Andy and Barney are standing beside the woman's car her rear-view mirror has vanished in the closeups, but when she drives off in the long shot the mirror has reappeared.
Visible crew/equipment: While Andy and Barney are standing beside Bobby Fleet's vehicle, Andy mentions that it's too bad they're all not staying overnight so they could hear Jim play, and when it cuts to the next shot we can see the reflection of the overhead boom pole and mic moving on the surface of the vehicle.
Visible crew/equipment: When Opie's up on the ladder feeding the birds, in the overhead shot of Andy and Barney the mic's cable running up under Barney's pant leg is visible.
Aunt Bee the Crusader - S4-E15
Revealing mistake: Toward the end of the show the rooster, Bo, is drunk. If you look closely you can see the two strings pulling the rooster's neck sideways.
Revealing mistake: During the holdup at the bank, while the bank robber's holding Asa's broken gun, when Andy motions to Asa to duck out of the way in the closeup we can see the two thin wires to the left and right pulling the gun apart.
Barney Mends a Broken Heart - S3-E6
Visible crew/equipment: While Andy and Barney are with Daphne and Skippy at the Tip-Top Cafe in Mount Pilot, when Daphne's boyfriend walks in we can see the actor's chalk T-mark on the floor.
Other mistake: Aunt Bee refers to losing the pickle contest 10 years in a row (which Clara corrects to 11, leading us to believe they have been in the same competition), but this is her 2nd year in Mayberry, having not seen Andy for years in her season 1 intro as the "new housekeeper", (or Opie - ever) in season 1.
Opie's Ill-Gotten Gain - S4-E8
Continuity mistake: When Barney is "reciting" the preamble to the Constitution, he messes up his hair. When they show him in close-ups it is messed up, but when it goes back to him and Andy in the scene it is combed. It does that a couple of times.
Mayberry Goes Hollywood - S1-E13
Visible crew/equipment: When Andy sees that several stores have made 'Hollywood' changes and then walks past the two men wearing suits, it cuts to Andy walking up to 'Hollywood funerals' and we can see the reflection of a crew member pushing the camera dolly with other crew members.
A Black Day for Mayberry - S4-E7
Continuity mistake: At the beginning of the show, two men walk into the courthouse. (As a sidenote, one of the men is Rance Howard, which is Ronnie Howard's real dad). Barney is the only one there, as Andy is out. The two men sit down to wait for Andy. Rance Howard is not wearing a hat, and he is in a light-colored suit. When Opie walks through the door looking for Andy, you can see that Rance is not sitting there any more, and now it's a man with a dark suit and wearing a hat. (00:01:55)
Revealing mistake: During the scenes with Opie and Dolly in Walt's barn, the horse is wearing a halter with a ring under the adjustable chinstrap, and in every closeup a thin wire (like picture wire) can be seen attached to the ring as it's being pulled from offscreen to have Dolly turn her head away from Opie. Then later when Walt returns and tries to feed Dolly, once again a thin wire attached to Dolly's halter is pulled from offscreen, so she will turn her head away from Walt. (00:05:55 - 00:18:20)
Visible crew/equipment: Just as the first shot opens of Andy walking down the block we can see the moving reflection of a crew member wearing a short sleeve, button down shirt, who's presumably sitting on the camera dolly as it rolls along keeping pace with Andy.
Barney and the Governor - S3-E15
Visible crew/equipment: After Barney has given the governor's chauffeur a parking ticket and the chauffeur drives off, just as Barney starts to say that he's not wearing the uniform to win friends, we can see the reflection of the boom mic at the top left corner of the screen.
Continuity mistake: After Eddie Brooke offers Barney five dollars and asks to be let go, it cuts to the State Police car passing the "drugs prescriptions" storefront, then Barney and Eddie trip and scuffle for two consecutive shots, and when it cuts back to the State Police car they haven't even passed that storefront yet. Additionally, that storefront is only two stores down from the coffee shop where Barney and Eddie are.
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.