At the beginning of the film when Teddy and Chuck are driven up to the asylum gate for the first time they are riding in a small open truck. On the rear of the vehicle a modern white license plate can be clearly seen as it reflects the light. The truck should have sported a black 1953 Mass plate like the one seen later on the doctor's red sedan.
A recording of Gustav Mahler's piano quartet plays in the SS officer's office at Dachau. Music by Jewish composers, like Mahler, was banned in Nazi Germany. No hard-line SS officer would have owned or listened to blacklisted Jewish music.
When the chief of the guards is giving Daniels his tour of the island, he says that the bluffs are tangles of poison ivy and live oak. The island is in Massachusetts but Southern Live Oaks don't grow north of southern Virginia.
After the storm blows out the electricity, Daniels comments about the "electronics" being knocked out. The electrical systems used to containing the patients likely had no "electronics", just electrical switches, solenoids, and other parts.
In Cawley's study, Daniels puts down his soda and ice on the table and emphatically shoves his hands in his pockets. In the few moments that the view changes to Cawley and Naehring and back again, and the glass is back in his hand.
Throughout the film there are many "jump cuts" - shots that don't match. For instance in the overhead shot of the Marshalls and the doctor in the cell, each shot shows them in a different position within the room. When Teddy is riding in the jeep with the uniformed officer, the close shots as they converse show Teddy looking in a different direction. Scorsese may have intended these for tension effect.
At the beginning of the movie when Teddy and Chuck are standing on the deck of the boat. Teddy lights a cigarette and clearly has a wedding ring on his finger. The next shot shows him with his left hand resting on the rail with no ring. The following shot once again he is wearing the ring and is doing so through the rest of the film.
During Teddy's dream Rachel is covered in blood and three dead children at her feet. She asks Teddy to give her a hand, Teddy approaches her and says, "I could get in trouble." In the shot from behind Rachel, she places her left hand on his right cheek. Then in the shot from behind Teddy her left hand is resting on his right shoulder.
When Teddy is stuffing his tie into the gas tank, he looks up to see his wife. He then holds the tie and balls it in his hands and says he's sorry for ruining the tie. In the next shot the tie is all the way in the tank. He wouldn't have started to put it in the tank, take it out, and put it back.
In a flashback, Teddy sees the entry gate of Dachau, with the words "Arbeit Macht Frei". It's actually the entry gate from Auschwitz, which was liberated by Soviet troops, and has the same words in a different style.
At Dachau, Teddy looks intensely at human bodies piled on top of each other, clearly frozen due to severe cold. Dachau however was not liberated during winter, but on April 29, 1945. It is true that some fresh snow had fallen overnight, but photographs taken by American soldiers show it was just a hint of snow, not a blanket. Late April wasn't cold enough.
Dr Crawley states that withdrawal from chlorpromazine causes tremor, which is commonly known as a side-effect of taking the drug. It is, however, also a symptom of stopping the medication suddenly, as are nausea, vomiting, dizziness, migraines, and hallucinations.
Towards the end when he runs into Dr. Naehring and takes his syringe you see him holding it by his shoulder, near the doctor's neck, from the shots behind him. When the camera is on his face, you can tell both of his arms are lower. It changes back and forth between shots.
When Teddy interrogates Bridget Kearns, Chuck brings her a glass of water, but when she is filmed drinking she has no glass in her hand. In the next scene she puts the empty glass on the table; however, she is putting the glass down with her left hand while she was using her right hand when pretending to drink. The imitation of drinking is likely to be intentional revealing of the "role play" performed for Teddy, than a mistake on the part of the actress, imitating drinking when there was a perfectly good glass on the table.
Teddy attempts to revive the children with CPR, which was officially "invented" in 1956, 2 years after the movie is set and 4 years after the children drown. However, CPR existed in many forms from as early as the mid-1700's. In August 1767 a few wealthy and civic-minded citizens in Amsterdam gathered to form the 'Society for Recovery of Drowned Persons', which within 4 years claimed that 150 persons were saved by their recommendations, including respirations in to the victim's mouth, either using a bellows or with a mouth-to-mouth method (mouth-to-mouth or mouth-to-nostril respiration is described including the advice that "a cloth or handkerchief may be used to render the operation less indelicate"). As a combat veteran it conceivable that Teddy witnessed such techniques.
When Rachel had blood all over her and three dead children at her feet, asking Teddy to give her a hand, the scene cuts a couple of times to Teddy's face with a cigarette in his mouth. These scenes are being played backwards, with smoke going into his cigarette rather than coming from it. This is a technique used by the filmmakers in editing for effect. It is deliberate and used to create a dreamlike atmosphere. Another use of reverse motion is seen when Dolores walks backward away from the lake in a flashback to the lake.
There are several clues and intentional continuity errors throughout the film that foreshadow the ending. This includes Bridget Kearns' drink of water disappearing between shots (she drinks from an empty hand), a cardigan appearing briefly on 'Rachel' (which is later worn by the other 'Rachel') and lines from Edward/Andrew's dreams being repeated (such as "Why are you wet, baby?"). These techniques are also used in his dreams showing a similarity between what he perceives as a dream and what he perceives as real.