When the Forty-Niner starts to cough, he moves his right hand to his mouth, but as the camera angle changes, his left hand is at his mouth.
When describing the as-yet-unseen aliens, Data refers to their Ophidian staff as being approximately 1.5 meters in length. When the rest of the crew are finally able to see the aliens for themselves, the staff is much less than 1 meter.
Data picks up a pair of glasses and identifies them as "bifocals", however the lenses are clearly flat, single-focal lenses and do not contain an additional bifocal area for reading or seeing near objects.
Data says the weapon found in the cave is a ".45 caliber double action cavalry pistol invented by Colt Firearms in 1873". The pistol Picard is holding, the one later used by Clemens, is actually a Colt Single Action Army, chambered in .45 Long Colt (was also originally available in .44-40 WCF and .38-40 WCF), which was made in 1873. Colt's first double action revolver was the Colt Lightning and wasn't released until 1877, and was not .45 caliber; it was only chambered for the .32 Long Colt, .38 Long Colt and the .41 Long Colt.
In the opening, Data identifies the revolver as a Double Action invented by Colt Firearms in 1873, but that should be a Single Action.
When Data offers his combadge for the ante in a poker game, he says that it is composed of "silicon, beryllium, carbon-70 and gold." Carbon isotopes have mass numbers from 10 to 15 but a mass number of 70 (with 64 neutrons) is not possible.
After discussing Data, Riker and Troi join Data in the turbolift, which continues on its way. However, neither of the two tells the turbolift where they want to go, as they ordinarily would, but exit with Data at his destination. While this is indeed an unusual procedure, it could be argued that Riker and Troi assumed correctly that Data was heading in their direction, i.e. the bridge, since all bridge officers had been called to their stations only moments earlier.
When Data looks at the newspaper advertising Guinan's literary reception, the writing on the back page of the paper in the next shot is back to front and upside down. This is because the newspaper has been folded in half to make for easier handling. Thus the back view is the bottom half of the page Data is reading and obviously it appears upside down..
At 34:03, after Data looks up from the newspaper, in the newspaper's upper left corner a stash of plain paper can be seen. It was used to make the newspaper appear thicker. Usually, on the set only the required page is printed, the rest are plain paper, but rarely visible.
Before the Devidians are seen, Data describes them as lacking apparent eyes. When the crew are finally able to view them, the performers' eyes can be seen through the effects used to realize their appearance.
Samuel Clemens is depicted with white hair and wearing a white suit, but this part of the show is set in 1893 when Clemens was 58 years old. According to the Mark Twain Boyhood Home and Museum, "In December, 1906, he wore a white suit while appearing before a congressional committee regarding copyright. He did this for dramatic emphasis. Several times after that he wore white out of season for effect. He has been portrayed as the white haired gentleman in the white suit, but this was only sporadically at the end of his life." Photographer Napoleon Sarony took several photos of Clemens in 1893; Clemens' hair was not all white in 1893, nor did Clemens look as old as he was portrayed in Time's Arrow.
When Jack gives a coin to the bookie and tells him to bet it on Gentleman Jim, the bookie responds with "Alright", but his lips do not move.
For a brief moment, a member of the production staff can be seen behind a wall in main engineering at the beginning of the show.
In order to get the main bridge crew to the past, they assemble a very illogical away team of just the series regulars. As a result, Counselor Troi and Dr Crusher help set up engineering equipment rather than the away team utilizing expert engineers. Despite the apparent danger of the mission, no security personnel apart from Worf beam down, either.
(9:00 minutes) When Troi and Riker are walking toward the turbolift, and waiting for the doors to open, a boom mic reflection can be seen above the turbolift doors as Troi & Riker are chatting.
When first arriving on Earth, Data & Picard are examining artifacts in the cave. Picard picks up a Colt revolver which Data incorrectly identifies as a Colt "double action" revolver when it is in fact a Colt "single action" revolver.
When Geordi and Data are in ten forward, Data tells Geordi he is rather comforting knowing that he's going to die someday. The feeling of comfort is an emotion, and does not belong to Data.
When Data is talking to Jack about the devise he is building he says the word "gonna" which is a contraction of "going to". Even though, he was repeating Jack's words, it is a running theme that Data cannot use verbal contractions. Even in the same sentence Data says "it is" instead of "it's" like Jack did.