It's Not Easy Being Green - S5-E16
Factual error: Boomhauer's car, Ms. Sally, has a Texas flag license plate on it. These plates were not issued until 1992, but Ms. Sally was driven into the quarry in the late 70's or early 80's.
Hank's Unmentionable Problem - S1-E6
Factual error: Hank is extremely constipated, but has a colonoscopy. He would not have been able to have a colonoscopy while extremely constipated, having not had a bowel movement in many days.
Factual error: When Nate goes to see Hank to show the incriminating photos of Bobby watering the lawn you can see Nate's car has orange tinting on his front headlight. Front headlights cannot contain any color tinting and must be clear. The car is an expensive looking car. I doubt he wanted to change the tinting of it either.
Returning Japanese (1) - S6-E21
Factual error: As the Hills get off the plane in Japan the camera crew is waiting for them as they exit the terminal however they do not go through any sort of customs area. Every international flight you must go through customs before entering the country.
Returning Japanese (1) - S6-E21
Factual error: During the scenes where the Hills arrive at the airport in Japan, there is no flight crew outside or around the terminal when they exit. Whilst they may be the last people to exit the plane, there will be one or more crew members to farewell passengers and also to close the terminal door. There doesn't seem to be any crew anywhere close, which seems particularly odd especially as Hank and Cotton manage to just stroll back onto the plane. A little after they leave the camera crew just go right through the terminal onto the plane without anyone there to stop them.
Returning Japanese (1) - S6-E21
Factual error: In the airport scene in Japan, the news crew arrives to film Cotton Hill. As Cotton Hill gets back on the plane trying to escape the news crew, we see a shot from outside the plane and the news crew is out on the tarmac. With security regulations at airports a film crew wouldn't be allowed to just go out onto the tarmac.
Three Coaches & a Bobby - S3-E12
Factual error: Before Hank knocks Coach Sauers out with the drink case, the coach starts up his car, but when Hank knocks him out the engine turns off as soon as Hank hits him. The engine wouldn't turn off because of being unconscious. (00:16:20)
Born Again on the Fourth of July - S13-E14
Factual error: When Bobby and Lucky first walk into the church service that is taking place inside the karate studio, the pastor is calling everyone a sinner, and proceeds to point to a picture of Bruce Lee behind him and says he was a big sinner who repented and knew that the unrighteous would not inherit the kingdom of Heaven. Bruce Lee was an atheist.
Take Me out of the Ball Game - S3-E24
Factual error: Near the end of the episode, Dale Gribble saves Team Strickland by catching a home run ball, hit by Kurt Bevacqua, with his hat. This wins them the game. However, according to the MLB Rulebook, if a player catches a fly ball with their hat, the batter takes three bases, meaning the game would continue and Bevacqua would be at third base. (00:21:00)
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.