Early on Hank states that he was born in 1953. Then in another show Hank says that he graduated Arlen High in 1974, which would make him born in 1956. Later Hank learns that he was born in New York on the day that Fidel Castro visited Yankee Stadium, which was April 15, 1959.
In earlier episodes, it's stated that Peggy grew up in Arlen and her mother was basically an older version of her. Later episodes change her backstory to say that she grew up in Montana, with a mother who was a harsh rancher.
In the early episodes, Luanne clearly has automotive skills - she is seen repairing cars and even ordering Hank parts he needs to fix his truck at one point. From "Jumpin' Crack Bass" on she has little to no knowledge of cars.
In "Peggy's Turtle Song", Tom Landry Middle School is stated to be in the "Heimlich County School District" according to the bus Bobby rides. But in later seasons the buses list it as the "Arlen Independent School District."
At the end the episode where Hank is having issues with Bowel movements, he rushes to the bathroom, and pulls Bobby out. Bobby is wearing PJ's and has toothpaste on his mouth. After Hank passes a movement, the shot is of Peggy and Bobby, and Bobby is now wearing his usual attire. When Hank emerges from the bathroom, Bobby again is wearing his PJ's.
In certain episodes, the Hill family goes to the hospital called Heimlich County Medical Center. In other episodes they go to the same hospital (according to looks) and the sign outside reads Arlen McMaynerbury Medical Center, not Heimlich County Medical Center.
Hank's drivers license lists his address in Arlen TX with a zip code of 73104. Although Arlen TX is a fictitious town, the zip code is real. However, the zip code they chose is in the wrong state - 73104 is in Oklahoma City OK.
In one episode, Hank explains that tow truck drivers don't need to have a Hazardous Material license to tow the propane trucks. In fact tow truck drivers are required to have all endorsements on their license in order to tow commercial vehicles.
The Returning Japanese two-parter states Hank inherited his narrow urethra from his father, while Cotton distinctly says in another episode he inherited it from his mother. Of course, Cotton says a lot of things, including the claim that he can pass a bowling ball through his urethra.