Arthur Graves's name and photograph are seen in the evidence chart in the lab before the team has actually identified him.
When Zack tells B&B to follow him to the endpoint of the laser trail, Booth bumps into the table supporting the emitter. The beam does not shift, but remains on its original target despite the fact that the source has shifted.
Both the judge and Caroline refer to Dr. Sweets as as psychiatrist when he is in fact a psychologist.
Booth is wearing a jacket and tie when Russ shows up in his office. A few minutes later when he takes Russ to the hospital he is wearing a T-shirt, jeans, and a green flannel overshirt.
When bones gets mad at booth he takes one hand out of his pocket but in the next scene he has both hands out.
Zack rigs a laser (or at least some kind of light-beam device) to bounce off the mirrors in the display. When he turns it on, the beam visibly travels to each mirror in turn (an order of magnitude of about 10 feet per second). However, light (laser or otherwise) travels at nearly 200,000 *miles* per second, or 7 orders of magnitude (10 million times) faster. Presumably, like Star Trek phaser beams, the error is deliberate because a traceable beam looks cool.
Hodgins describes the original Gormogons as a 17th century cult. The historic Gormogons were actually an early 18th century group, if they existed at all. They may have just been an elaborate practical joke by some anti-Masonic gentlemen. Furthermore, the Freemasonry movement formalised in early 18th century, and Illuminati (both Bavarian and Avignon orders) have been founded in the last quarter of 18th century.
Hayley's monitor screen shows a computer-graphic display for mechanical ventilation, yet she has only a nasal cannula for oxygen.
At the mausoleum, Booth reads the Latin words "Dead Father", saying he was an altar boy. But in Season 1 he can't read the Latin phrase "I carry with me all my things". However, this could easily be due to a knowledge of some simple Latin commonly used in a church service (with both "dead" and "father" being such words).
Brennan & Booth assure the Archbishop several times that only a part of the victim's body was cannibalized (the knee caps to be added to the silver skeleton), yet in "Bones" (2005) {The Widow's Son in the Windshield (#3.1)}, the skull of the violinist was eaten, even though only his left pinkie finger was used in the skeleton. However they don't directly say that just the victim's kneecaps were eaten, only that part of him was.
Booth claims to know Latin because he was an altar boy. It is true that Catholic church largely discontinued Latin masses in 1965 (i.e. at least a decade before Booth could have become an altar boy) and most altar boys did not learn the Latin language but rather memorized the prayers by rote, but it has been stated before that Booth does not know Latin. His knowledge of few words (especially those similar to their English counterparts) could have been acquired incidentally.
Right after booth and bones get blown up when booth says the corrupter as being the next target his lips do not move.
The envelope containing the kneecaps state that Dr. Brennan's apartment is in the northwest part of DC. However the zip code on the envelope is in the southeast part of DC.
In the scene with the car tailing, a street sign can be seen which is in the style of Los Angeles, California signs (as opposed to in the style used in DC and Maryland, where the scene takes place).
When Hodgins is explaining about the apprentice and masters, he is pointing to a board behind him. On the board is a picture of Arthur Graves, the gormagon master they find in the nursing home later in the episode.
Zack rigs a laser to show Booth and Brennen the mirrors that lead to the camera. The vault was moved from the original bank to the basement at the Jeffersonian Museum. The chances of them setting every piece exactly how it was in the bank vault so that the mirrors would line up perfectly are slim to none.
When driving the taxi, Booth doesn't appear to be wearing his seat belt. After the explosion, the car does a complete flip. Afterwards, Booth is sitting perfectly in his seat, and only has minor injuries. He even proceeds to run off and chase the apprentice around right after. Since he wasn't wearing a seat belt, he should have been thrown around when the car flipped and had much worse injuries, if not died in the crash.
Booth calls Archbishop 'Your Holiness' which is a title reserved for the Pope. No observant Catholic would have made such a grave mistake.