Peggy Hill: the Decline and Fall (2) - S4-E1
Continuity mistake: Before Hank starts arguing with Cotton near the end of the episode, Cotton has a can of beer in his hand. When Cotton is being held by Hank and swinging his arms at Hank the can of beer has disappeared. There was no moment where we can hear him drop the can, and as he pushes Hank we never see him drop it.
Peggy Hill: the Decline and Fall (2) - S4-E1
Continuity mistake: When Peggy is drinking the formula for breakfast, she has a small napkin placed over her neck cast. As Peggy and the baby start crying the napkin has disappeared.
Continuity mistake: After Peggy has fallen out of her wheelchair she had her bag on the back of her wheelchair. When Cotton is taunting her and pushes the chair near her, the bag has disappeared.
Bills are Made to be Broken - S4-E3
Continuity mistake: Whilst Bill is practising football in Hank's yard there is an aerial shot that shows Bobby & Hank walking towards Dale & Boomhauer and are close to them. In the following shot they are quite a bit further away from their previous position. Also the position of Dale & Boomhauer next to the garage changes between shots.
Bills are Made to be Broken - S4-E3
Continuity mistake: When Hank and Bill are sitting on the hood of Bill's car and talking, one shot of the football field shows it to only be about 70 yards long between the end zones.
Little Horrors of Shop - S4-E4
Continuity mistake: When Dale & Hank are taking a spa in the house Dale is fumigating, Dale gets out of the spa. We see Hank has a headrest behind his head that is pink. The next shot is a close up of Hank's face and the headrest is now a light brown.
Little Horrors of Shop - S4-E4
Continuity mistake: Later in the episode, the students go over to Hank's garage to finish some of their class projects and gather around the workbench about halfway through the scene. Between shots some of the kids change positions including Joseph and one of the kids in a pink shirt.
Little Horrors of Shop - S4-E4
Continuity mistake: After Hank has been made to take forced leave from work, there is a shot of the house and the background you can tell by the color the sun is rising. In the next shot when Peggy is pretending to conduct in the kitchen with the kitchen utensil you can see a blue sky outside the kitchen window. There is no way the sun rose that quickly in the space of a second.
Little Horrors of Shop - S4-E4
Continuity mistake: During the scene where Peggy is taping a photo of Hank to the blackboard to discuss about the substitute teacher of the year. When she tapes it up you can see it's near the top of the board near the writing accompanying it. After a shot of the students the photo has moved down quite a bit along with the writing that was there.
Continuity mistake: When Hank takes Connie to the doctors, he is in the waiting room. After Connie and the doctor come out they open the two doors and stand in front of the doors. Towards the end of the scene as the doctor is about to leave them, they have all moved far a short distance from the left of the doors.
Continuity mistake: After Connie tells Hank she is experiencing her first period, Hank consults the book that Connie's parents left for Hank. There is a close up of the folder from Hank's point of view and you see there are not many pages on each side. You can tell this as you can see the binder's covers below the papers on each side and there is not much material on either side. The next shot that shows Hank skimming through the papers each side of the folder has a lot more papers on each side than the previous shot.
A Beer Can Named Desire - S4-E6
Continuity mistake: When Dale is meeting Joubert, you can see Lilly (blonde girl)is to his right. Next shot, she is on the other side of Joubert and next to the other girls.
A Beer Can Named Desire - S4-E6
Continuity mistake: After Violet has introduced herself to Bill you can see she has Joubert & Lilly to her right and in front of Bill. The following shot she has moved away from Joubert and is next to Bill's right side.
A Beer Can Named Desire - S4-E6
Continuity mistake: Toward the end of the episode Don Meredith takes Hank to the Superdome to try throwing the football into the Alamo beer can. After Hank makes the throw they walk to the football rack and Don picks up one of the footballs from the rack. We can see the top shelf has two footballs with the laces on the ball not visible. As they leave we can see on the same angle, there are three balls on the top shelf with the laces visible on all three balls.
A Beer Can Named Desire - S4-E6
Continuity mistake: As Hank is discussing the competition with Peggy he is wearing his light blue boxer shorts. After Peggy asks whether he is going to still enter the competition, they are now dark blue.
A Beer Can Named Desire - S4-E6
Continuity mistake: After Bill kisses Rose on the tombstone you can see he has quite a few sweat stains visible on his shirt. Bill then goes inside the house looking for Rose and the stains have disappeared.
Continuity mistake: As Hank is calling Peggy's mother to tell about the delay there is a sign above the telephones that says "TELEPHONES". When he hangs up the sign now says "TELEPHONE".
Continuity mistake: Luanne in one scene tells Hank a man asked if she would like to join the mile high club and when she tells him she is facing Hank. In the next shot she is standing in the same position but has turned around and is facing away. There was not enough time for her to move between shots.
Continuity mistake: Whilst the guys are waiting in the alley for Hank to get back (before Boomhauer leaves) you can see the backhoe is behind Boomhauer. When it changes to showing the empty street from their point of view, the backhoe is now down the street. The following shot its back to behind Boomhauer.
Answer: In addition to Brian Katcher's response: it's supposed to be a joke that Dale is always suspicious about conspiracy theories, mysteries, etc, but he is "clueless" about his own wife having an affair. Of course, as Brian wrote, many fans think that Dale is pretending to be clueless because he doesn't want Nancy to leave him.