When Will and Sean are discussing Will's record and how they were both physically abused as kids, Sean's cardigan sweater goes from hanging near his shoulders to being pulled up with his shirt collar tucked underneath and back again.
When in court for his assault on a police officer, Will begins to quote Henry Ward Beecher's "Prayers from the Plymouth Pulpit" and dates it to 1887. It was actually published in 1867. Henry Ward Beecher died in 1887.
When Will and Skylar are at the outdoor cafe near Harvard Square talking about the complicity of Organic Chemistry, her front shots show her hair on her shoulders, and the rear shots show her hair in the middle of her back.
When Will burns his proof of a problem and Gerald puts it out, the amount of burn damage changes from almost completely destroyed (when Gerald first extinguishes the flames) to minimally burnt (near the end of the scene), and various other degrees in between.
When Sean is packing his suitcase at the end of the movie, the window to his left has the curtains closed. But in the very next shot, the curtains are open as the camera is looking at Sean's left side.
When Chuckie is having the conversation with Will about how he wishes one day Will wouldn't be home when he comes to pick him up, he finishes the scene with a sip of beer from a can. The can is in his right hand as he takes the sip, but in the next shot as he lowers the can from his lips, it has miraculously jumped to his left hand.
At the end after Will leaves and Chuckie goes to pick him up and realizes he's not there, while he's walking back to the car he tells the other guys that Will's not there. However, it can be seen from his lip movements that this is not what he actually says.
Near the ending during the scene where Chuckie is at Will's house to pick him up, as Morgan gets out of the back seat of the car to change and sit in Will's old seat, while he's climbing into the passenger seat at the bottom left hand corner of the screen you can see a crew member with a red hat on watching from the sidelines.
When Professor Lambeau is interrupted at the alumni cocktail reception, the alumni wearing the red jackets are all too young to be wearing the he "Artillery" Red jacket, which is the MIT 50-year alumnus jacket.
The U.S. Army general who sits next to will during his interview at the NSA has several things wrong with his uniform; he's missing the thick black band that generals wear on their lower sleeve, he has some non-US decoration under his name badge that shouldn't be there, he wears Infantry branch insignia (crossed rifles) on his lapels which a general shouldn't wear and his cap has a braided 'chin strap' (above the visor/peak) which is wrong for any US army cap.
When Will is seen driving to Stanford during the closing credits, the sign on a bridge over the highway says "Prospect St., Stockbridge", which points to him being on Rt. 90 west (MassPike). This means Will was taking an 18-mile/1-hour detour through upstate NY and Cleveland, Ohio, as the faster way to go (according to Google Maps) is via Rt. 84 south through Connecticut.
However, prior to Google Maps (and indeed even today), most people driving from Boston to San Francisco take the direct route of I-90 to Chicago and then I-80 from there. In spite of Google Maps, due to higher speed limits, Will's route is actually the faster route.
While at the race track, Skylar, while rooting on her dog, mentions that it is winning the race. However, the camera cuts to the race and the "2" dog overtakes the leader and wins. She then states how happy she is that she won.
Earlier in the film, Chuckie expresses his wish to come to Will's door, to find that he has left without notice. This happens at the end, however, the group has already given Will his car, eliminating the need for Chuckie to come pick him up.
When Sean tells Will about how his wife used to fart all the time, even in her sleep, everybody is laughing. Even the cameraman. The camera starts bobbing up and down at two very funny moments during that dialogue.
In the scene where Skylar and Will are sitting at a table with the chess players behind them, the white noise in the background of the shot lessens for Will's line "and I can't play the piano" and the intensifies again afterwards. If you listen closely, you can tell that the one line is a voice over.
At the end of the movie when Chuckie goes to pick up Will, he's wearing short sleeves and shorts. It's probably August or September and the trees still have their leaves. However, it's cold enough that you can see Chuckie's breath several times.