Goober Takes a Car Apart - S5-E17
Factual error: Halfway through the show, Goober takes the motor out of the car and they start it up in the Sheriff's office. If you look at the direction of the fan, it would be blowing the air forward instead of towards the rear of the engine. It would be fighting the air coming through the radiator instead of pulling it.
Goober Takes a Car Apart - S5-E17
Factual error: How in the world did Goober lift that heavy motor out of the car?
Visible crew/equipment: While Andy's sitting on his desk going through the mail and Barney's sitting on the side desk, the actors' chalk marks are visible on the floor behind them, in the shots facing both of them.
Barney Runs for Sheriff - S5-E21
Other mistake: When Floyd tells Barney that he doesn't want to put Barney's campaign poster in the barbershop window because it spoils the decor, the reflection we see in the mirror behind Floyd is the church's front door and steps. This makes no sense at all because the reflection should be whatever is to the right of Floyd's, but the church is the opposite way - to the left, past the courthouse. The view in the mirror is a painted backdrop, but it's still quite wrong.
If I Had a Quarter Million Dollars - S5-E22
Continuity mistake: After Barney makes his way around town spreading The Rumor that he has loads of money, when Barney walks into the courthouse he removes his jacket and hat then hangs them on the coat tree, and the hat keeps moving under and over the jacket between shots.
Continuity mistake: In the exterior shots of the bank's doors there are rectangular door knob backplates on the outside of the door, but in the interior shots when the door's open we can see the door knob's backplate is narrower and has a decorative top.
Continuity mistake: When Barney walks into the bank and turns on the light, we can see there's no damage to the side of the door, but when Andy walks in he points out the area where the door was jimmied.
Continuity mistake: When Barney's outside the bank loading his gun with the bullet, there are window blinds on the inside of the doors' windows - nothing else, but when Barney's inside the bank there are white shades between the glass panes and the blinds. As for the bank's large side window the opposite happens - there's a white shade between the pane of glass and blinds in the exterior shot, but in the interior shots we can see through the blinds' slats and there's no shade.
Other mistake: When Andy and Helen are in the squad car alone after going to the movies with Gloria, both car doors are closed, but in the closeups from outside both sides of the car, note that both triangular vent windows are missing and we can see the doors' rubber seal - both doors are open for these closeups.
Visible crew/equipment: When Gloria and Frank drive off at the end, just as the camera zooms in on Andy and Helen, the shadow of the moving boom mic is visible at the top, left side of the screen.
The Case of the Punch in the Nose - S5-E25
Other mistake: In the shots from outside the courthouse, when the front door is opened there's a wall where none should be, presumably to prevent seeing inside the courthouse facade.
Other mistake: After the preacher walks past Andy and Sam, when the two men comment that the preacher's sermons "go on" and are "dry as dust," we can tell that all of the store's glass window panes have been removed, presumably to avoid the crew/camera's reflections as the shot pans across the window to Howie.
Aunt Bee's Invisible Beau - S5-E27
Revealing mistake: When Clara and Aunt Bee are talking in the kitchen, in the closeups facing Aunt Bee we see the counter, the cabinet, and the wall, but what we don't see are the curtains, the window, or the windowsill, which all should be on that wall. The fake window wasn't placed on the set wall for Bee's closeup.
Aunt Bee's Invisible Beau - S5-E27
Other mistake: After Andy talks to the butter and egg man in Mt. Pilot, when Andy gets in the squad car with Barney all the car doors are closed, but in the closeups from outside both sides of the car, note that both triangular vent windows are missing - the two doors are actually open for these closeups.
The Arrest of the Fun Girls - S5-E28
Continuity mistake: When Helen and Thelma Lou walk into the courthouse and find Andy and Barney dancing with Daphne and Skippy, there are lipstick marks on Barney's cheek, but when Andy gives Skippy her sweater and radio then tells them both to go, in the closeups the lipstick marks are gone, though they reappear momentarily.
The Arrest of the Fun Girls - S5-E28
Continuity mistake: When Daphne and Skippy drop in on Andy and Barney at the courthouse and want to "play jail," Skippy carries the radio and her sweater into the jail cell with her, but when Skippy asks for her ukulele we see the radio and sweater are now on the side desk, under the gun case.
The Arrest of the Fun Girls - S5-E28
Other mistake: When Andy, Helen, and Thelma Lou are parked in front of the courthouse the lights turn on, and in the closeups from outside both sides of the car, note that both triangular vent windows are missing - the two doors are actually open for these closeups. This happens again later, when Andy and Barney are each apologizing individually to Helen and Thelma Lou.
The Arrest of the Fun Girls - S5-E28
Character mistake: Barney states when the Fun Girls were in town before, in Season 4, Thelma Lou didn't speak to him for 10 days. That's not true, they made up the same day.
Opie Flunks Arithmetic - S5-E30
Continuity mistake: At the end, when Helen shows up at the courthouse and tells Andy that Opie got a B+ on the test, her purse vanishes and then reappears while she's sitting on the banister beside the desk.
Opie Flunks Arithmetic - S5-E30
Continuity mistake: When Barney, Andy, and Aunt Bee are at the table after Opie goes upstairs to study, in shots that face all three of them there are things on the middle shelf to the left and right of the center plate, but then in other shots either the left item is gone, or the right item is gone.
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.