Visible crew/equipment: At the end, when Barney's talking to all the boys at the courthouse, the camera angle's looking up in Barney's closeups, and the top of the courthouse set is visible right before Barney mentions "intimidation."
Visible crew/equipment: After Barney brings the "Fun Girls" to Andy's house, when Aunt Bee walks into the parlor there are now three black tape marks visible on the carpet - and those marks weren't there when Barney and the women walked in.
Continuity mistake: While Aunt Bee and the party guests are gathered in the parlor, when Gomer hoots like an owl in the shot facing him note Floyd's eyeglasses - black with thick straight temples, but in the shot from behind Gomer and all following shots Floyd's wearing different eyeglasses - lighter color with curved tips.
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.
Other mistake: When Barney, Thelma Lou, Helen, and Andy are in the squad car talking about getting dinner, there's black tape on the chrome trim of the triangular vent window - presumably to prevent the camera/crew's reflections; and the vent window is shut in wide shots, but open in the closeup - also to prevent reflections.
Continuity mistake: While Barney and Thelma Lou are stalling Andy and Helen in the squad car, note the long whip antenna is missing (the empty bracket that always holds the antenna in place is over the rear passenger door) when they drive off to Andy's, but when they pull up to the house the antenna's on properly.
Visible crew/equipment: At the courthouse, while Barney and Thelma Lou are trying to stall for time, in the shots facing Andy and Helen when Andy says, "Steaks, chops, macaroni surprise," we can see the shadow of the boom mic moving around at the top, left side of the screen.
Revealing mistake: When Andy, Barney, Gomer, and the kids are at the campsite in the woods, during the daytime scenes the only shadows that should be seen are the ones being cast by the sun, but there are shadows being cast in different directions due to the set lighting.
Deliberate mistake: Gomer leaves the woods after they were lost and doesn't have anything with them then when they get to the campsite he has the snare that they made.
Continuity mistake: When Andy's talking to Gomer through the window while Gomer's holding the pail, in the shots facing Andy the pail's handle is facing out, but in shots facing Gomer the handle is facing the opposite way.
Revealing mistake: At the end, in the shots of the kids dancing in class, the width of the wall between the shelf/coat hooks and the perpendicular wall with the door has bizarrely increased by 1-2 feet, but that extra width is gone in previous and following shots. Presumably they needed extra space for all the kids to dance so they moved that left wall.
Other mistake: In this episode (aired Sep. '64) Barney tells Andy that 5 years ago on May 16 (1959) he joined the sheriff's office, and later they all have a surprise party for Barney. However, in season 4 "Citizen's Arrest" (aired Dec. '63) Andy tells Barney that he found, "The custody receipt for the first revolver I issued you... August, 1953... 10 years ago last month since I swore you in." Andy even mentions having a party with a cake that has 10 candles on it.
Visible crew/equipment: While Ollie's making up excuses to rush home, he and Nora are saying goodbye to Aunt Bee, and the reflection of the moving boom pole and mic are visible on the car's windshield.
Continuity mistake: When Aunt Bee, Andy, Ollie, and Nora are in the kitchen for breakfast, the bag of flour vanishes from the counter depending on the angle of the shot.
The Education of Ernest T. Bass - S5-E4
Other mistake: After Ernest T. breaks Helen's window Andy shows up, and there's black tape adhered to the chrome trim at the side of the windshield and also the chrome trim beside the vent window.
The Education of Ernest T. Bass - S5-E4
Visible crew/equipment: When Ernest T. runs across the street to help Helen with her groceries she rushes away from him and heads up the block to Andy, and as Andy and Helen walk toward the courthouse the shadow of the boom mic passes across the white pillar in front of the door.
The Education of Ernest T. Bass - S5-E4
Continuity mistake: In Season 3, Mountain Wedding, Earnest T knows how to read and write. He writes a letter to Andy and uses a rock to throw it in. In The Education of Ernest T Bass, we are told that Ernest T cannot read nor write.
Continuity mistake: Barney tries showing Andy what the "part bloodhound" dog can do for The Manhunt so Barney blows a dog whistle, and in the closeups of the growling dog the rope that was tied to his collar is suddenly lying on the floor, but it's back on his collar in the following shot.
Revealing mistake: When Barney blows the dog whistle to call Blue from the back room, the dog comes running and knocks Barney on his back, and we can see a bit of the cushion that's under Barney's back - it's visible within the shadow under him.
Other mistake: When the escaped convict takes Barney hostage, they walk away from the lake and Barney drops the piece of paper with the convict's picture. Later, when Andy walks across the grass he finds Barney's cap and the paper lying on the grass, but the problem is that Andy needs to pull up at the center of the paper because it's pinned to ground, presumably so it wouldn't move or blow away for the shot.
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.