The Big Bang Theory is a perfect example of a show that exploits popular notions about nerd culture and does its best to exaggerate it to a point where it seems funny. Most of the jokes show us (the real nerds) in bad light but it's all cool as long as they have Howard and Raj in the show. The Big Bang Theory has a number of recurring gags that make a lot of situations funny and memorable. The show runners take these gags pretty seriously. Some of these gags are inaccurate and have inconsistencies.
Here are a few mistakes of the popular show that render a lot of situations and quirks useless -
1. Sheldon's Spot
Sheldon is very particular about sitting in his spot but he has no issues with Leonard sitting there.
Dr. Sheldon Cooper has been very clear about his love for his spot. Sheldon has placed this location "in a state of eternal dibs", and has little tolerance for anybody who tries to sit there. Sheldon says
"In an ever-changing world it is a single point of consistency. If my life were expressed as a function on a four-dimensional Cartesian coordinate system, that spot, at the moment I first sat on it, would be (0,0,0,0)."
In The Bat Jar Conjecture in Season One, ahead of the Physics Bowl, Leonard is allowed to sit in Sheldon's spot without any trouble. Even at the start of Season Two, Raj sits in Sheldon's spot without facing Sheldon's wrath. Even Howard sits there in one of the episodes.
2. The Broken Lift
There's a contradiction in the show's timeline that shows that the lift was broken twice.
The elevator has remained broken throughout the show. We've had a lot of jokes when the characters were moving down the stairs. This running gag is rendered pointless when you carefully look at everything that involves the elevator.
In The Nerdvana Annihilation in Season 1, when the gang buys the time machine, Leonard says the lift has been broken for two years, but then in The Staircase Implementation in season 3, it is established that Leonard's failed experiment blew up and broke the lift seven years ago. Later on in the show, Howard also tries to figure out what is wrong with the lift and then ultimately gives up. Why would he do that since he was already present when Leonard blew up the lift?
3. Sheldon is a genius who is wrong a lot of times
Not to be cherrypicking the mistakes here but a show based on the brilliance of Sheldon shouldn't have the character being wrong so many times.
Sheldon is established as a genius pretty early on in the show. The show becomes a lot about how this brilliant scientist has to deal with the other guys who aren't as smart as him. He claims (with Amy) that Danishes wouldn't exist if Kopenhagen were flooded, despite the pastry being invented in Vienna; that ears don't cross hemispheres (they do) and claims the sound produced by hitting a wine glass to get everyone's attention is a B Flat, when it is in fact a B. These issues become annoying later on as he starts being stupid on different occasions. Sheldon even got the spelling of "fusion" wrong on one of his white-boards.
4. Sheldon's weekly schedule is all messed up
Sheldon is a creature of habit but his weekly schedule is often ignored.
In The Dumpling Paradox, Penny sleeps on Sheldon's couch thus preventing the latter from enjoying Doctor Who's morning run on a Saturday. Going by that logic, the previous night should have been a Friday but we see in the show that the previous night was Halo night. There's a running gag in the show that Wednesday nights are Halo nights. So was it a Wednesday or a Friday? This weekly schedule makes no sense.
5. Sheldon is allergic to cats and must never be able to adopt them
Sheldon is said to be allergic to cats early on in the show but then goes on to adopt cats in The Zazzy Substitution.
During the early seasons of the show, it is established that Dr. Sheldon Cooper is allergic to cats. In the episode, The Zazzy Substitution, Sheldon adopts six cats - Dr. Robert Oppenheimer, Enrico Fermi, Richard Feynman, Edward Teller, Otto Frisch, and Zazzles. If he was indeed allergic to cats, Sheldon should have been dead by now.
6. Leonard's relationship with Kim Joyce lasted for 27 days and not 12 hours
Leonard's relationship with Kim Joyce is said to have lasted 12 hours but earlier it was mentioned that they both dated for 27 days.
In Season 2 of The Big Bang Theory, when Leonard is dating Stephanie; Sheldon, Raj and Leonard discuss his previous relationship with Kim Joyce. They mention the fact that their relationship only lasted 27 days. In the Season 3 episode, The Staircase Implementation, it is shown that Leonard slept with Kim within 12 hours of meeting her. Sheldon's roommate agreement prevents Leonard from having sex with her and it basically drives her away. So was his relationship a 12 hour long fling or a 27 day long relationship?
7. Does Buffy - The Vampire Slayer exist in their world or not
Eliza Dushku (a star from Buffy) appears in the show as a FBI agent but the gang doesn't recognise her at all.
Eliza Dushku was cast as a FBI agent to appeal to the numerous nerds watching the show. In the episode The Apology Insufficiency, Eliza interrogates the four guys. Eliza Dushku is a star from the popular franchise Buffy the Vampire Slayer. Howard, Leonard, Raj and Sheldon are shown to be fans of the franchise but how do they not recognise her as a FBI agent? If they love the show, they must recognise her. This doesn't add up well with a show that has the main characters recognise real actors from other pop-culture entities.
Here's Eliza Dushku in Buffy - The Vampire Slayer
8. Penny doesn't remember her father's name
In The Maternal Congruence, Penny says her father's name is Bob. But when he is introduced in the show, his name is Wyatt.
One of the major criticisms of the show during its initial seasons was the fact that Penny was such an expendable character that even her last name was never mentioned in the show. In The Maternal Congruence, Penny says her father's name is Bob. This is the first time she mentions her father's name. Later on in the show, Penny's father makes an appearance and is referred to as Wyatt. Since their surnames aren't mentioned, her father's name is definitely not Bob Wyatt. She definitely has some daddy issues if she can't even remember her dad's name.
9. Sheldon saved a nun by giving her CPR
Sheldon wouldn't give a CPR to a complete strange because of his severe germaphobia.
Sheldon talks about giving the kiss of life to a nun on a bus as the only time he has kissed, but I can't imagine with his germaphobia how he would ever have done that. This is completely bizarre how Sheldon would overcome his fear of germs for a stranger. This doesn't go well with the personality of Dr. Sheldon Cooper.
10. Sheldon's weight keeps changing in the show
Dr. Sheldon Cooper keeps changing his body weight without any change in the body shape.
Sheldon's weight fluctuates without him actually changing his body size. When driving with Penny to the grocery store in The Luminous Fish Effect, he states he is 140lbs, but in The Porkchop Indeterminacy, he tells Missy that he weighs 165lbs yet with that 25lb gain his body is exactly the same. This isn't really possible because there would a significant change in his body if he were to gain 25lbs.
11. We actually see Howard Wolowitz's mother a lot of times
Howard Wolowitz's mother is one of the key figures of The Big Bang Theory. A lot of people often believe that they have never seen her but that's not the case. She isn't as fat as the stories make her out to be.
Right from the start of the show, we hear about Howard's mom in a not-so-great way. She is referred to as fat and ugly. We never really see her in the show as well. This turned into a recurring gag on the show. We actually see her thrice in the show - during Howard's marriage, when Raj was over for dinner at Howard's house and during a Howard Wolowitz monologue.