Continuity mistake: After Andy and Floyd talk outside the grocery Floyd gets into his car with the convict, and when they drive away we see all the food ads taped to both of the store windows while Andy's on the sidewalk, but when it cuts to Andy's closeups the taped ads have changed.
Other mistake: Floyd's car has the license plate number "RD 757" which has been seen on other vehicles in previous episodes.
Character mistake: In the first scene of the show, Andy, Opie and the mayor are on a pier fishing a lake. They are casting with deep sea reels. If you've ever tried casting with a deep sea reel, it doesn't work out well. You can see there no equipment on the poles.
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.
Visible crew/equipment: When Barney walks into the bank dressed as a woman, we can see the two curved chalk actor's marks on the floor behind the teller.
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.
Barney and the Governor - S3-E15
Deliberate mistake: In the closeup of the license plate on the governor's car, the words over and under the plate number are deliberately taped to be unrecognizable. The original words are "drive safely" and "North Carolina."
High Noon in Mayberry - S3-E17
Deliberate mistake: While Luke and Andy are in the house, Barney, Gomer, and Otis are hiding on the porch, and Barney has the idea to turn off the lights in the house, so he goes to the circuit box located beside the windows on the front porch and switches the lights off. The problem is that this circuit box is located on the porch only for this episode as a plot point.
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.
Factual error: We know from 2 episodes prior (Man in a Hurry) and generally mentioned throughout the show, the only filling station in Mayberry is Wally's Filling Station. In this episode Mayor Stoner is accused of only putting the new underpass in Mayberry because it will be next to Mayor Stoner's brother's filling station.
Continuity mistake: In the closeup of Barney's yearbook picture his name is listed as "Bernard Milton Fife," and Andy even reads it aloud, but in the previous episode from season 1, "A Plaque for Mayberry" Andy says that Barney's middle name is Oliver.
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.
The Great Filling Station Robbery - S3-E22
Visible crew/equipment: When Gomer walks into the courthouse with the envelope and Barney takes the photo out, the shadow of the boom pole is moving around on the door and wall to the right.
Aunt Bee's Medicine Man - S3-E24
Continuity mistake: The pillow on the right side of the couch is there while Opie is talking to the Colonel. When Andy walks around after Opie leaves, it has suddenly disappeared. (00:16:00 - 00:18:00)
Aunt Bee's Medicine Man - S3-E24
Continuity mistake: Andy and Barney are playing checkers near the end of the episode. Andy can be seen making 2 moves in a row, skipping Barney's move.
The Darlings Are Coming - S3-E25
Visible crew/equipment: During the scene where the Darling's truck has overheated, Andy walks up from the back of the truck. As he comes into view you can see what appears to be a microphone cord hanging out from under his pants leg.
The Darlings Are Coming - S3-E25
Visible crew/equipment: After Charlene tells her father that she just wants to look at the pretty man (Andy), when she gets back in the pickup truck we can see the reflection of the moving boom on the windshield.
The Darlings Are Coming - S3-E25
Continuity mistake: When Briscoe gets into his truck, his hat is wet from dipping it (5x) in a horse trough. The camera cuts away to Andy and immediately back to Briscoe, whose hat is now dry. Also, "she's always hood for 11 hat fulls," yet Briscoe only puts 5 hats of water in his engine.
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)
Visible crew/equipment: While Andy and Barney are hiding behind the car, when Mrs. Lesh hits Jake in the stomach with her umbrella, we can see two curved chalk actor's marks and two chalk T-marks on the floor. (00:22:10)
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.