Character mistake: Game Over: In the flashback of God playing the Simpsons Game, there is a newspaper article on the wall with the headline "MAN ON MOON." Naturally, there will be spaces in between each of the words, but there are also spaces in between some of the letters in the word MOON; there is a space in between the M and first O, and a space in between the first and second O that shouldn't be there.
Character mistake: Bargain Bin: When Bart is researching the Simpsons Game online, he says "Bartman, eating contest, alien invasion, slaughtering dolphins, it's describing everything we've done all day." Even if we were to assume the events of Bartman Begins through Mob Rules took place over one day (which is already hard to believe), the family were shown sitting at the breakfast table in the morning during the Mob Rules end cutscene, so there is no way the entire game took place over the course of one day.
Character mistake: Invasion of the Yokel Snatchers: When Homer and Bart witness Cletus getting abducted, Bart tells Homer "they're torturing that yokel", almost as if Bart doesn't know who he is. The Simpson family has interacted enough times with him throughout the series to know who he is and that his name is Cletus.
Character mistake: Game Over: After returning from the Game Engine and Bart points out they are still under attack, Marge says "I guess all that's left to do is pray that God saves us." When she says this, Lisa can be seen smiling, but the town is still under alien invasion so there should be no reason to do so. She doesn't come up with the idea to use her Buddha Hand powers to talk to God until right after Marge says the line, so it couldn't have had anything to do with that.
Character mistake: Mob Rules: Marge is reading a newspaper regarding senators trying to push Mayor Quimby to ban Grand Theft Scratchy. She looks at the bottom of the page and discovers Quimby is cosying up to a sleazy video game producer; she disapproves of this, but she appears to be smiling when saying "That really burns my beehive" despite there being no reason for her to do so.
Character mistake: Big Super Happy Fun Fun Game: During the battle between Homer and Lisa and Sherri and Terri (who are both female), just as Lisa unleashes Linguo, Lisa says "Linguo, fix his grammar for good." In this context, it should be "her" or "their."
Character mistake: In the Simpson house, the time on the grandfather clock in the hallway reads 12:15. But the clock in the kitchen reads 10:00.
