Factual error: Bone mentions Booth holds the "official record for the longest shot, almost a kilometer", and Booth corrects her to "over a kilometer." The record at the time was actually well over 2km, held by Corporal Rob Furlong, a Canadian sniper (since been beaten, it currently stands at 3.5km). In fact plenty of sniper kills had been recorded at over a kilometer since the late 19th century.
Continuity mistake: In this episode, Booth and Bones are telling Sweets about their "real first case" together - the case where they actually met for the first time. During this episode, Booth, Bones, and Angela take some evidence to Caroline, and Booth introduces them both (most importantly Brennan) to Caroline. However, in season 1, episode 19, "The Man in the Morgue", (which would have taken place after this initial first meeting) when Brennan is accused of murder in New Orleans, Booth has Caroline fly to NOLA to be Brennan's lawyer, and he introduces the two of them. If they had already been introduced during the first case Booth and Brennan worked on together, they would not have needed to be introduced here; they would have already known one another.
Factual error: The Roswell, New Mexico sheriff forces Bones to examine the body because it was found in his jurisdiction. But later in the episode, we learn that the body was found 2 kilometers from the border of Mexico. Roswell is not even in a county on the border of New Mexico much less one that borders Mexico. In fact the closest border to Roswell between the US and Mexico would be in Texas - hours drive away. The Roswell sheriff would have no jurisdiction.
Character mistake: Hodgins is talking to Bones and says something about Goodman not being able to see his family and Zach not being able to see his kids. Goodman has kids, Zach was going to visit his family, not the other way around.
Plot hole: At first the victim (Careful Lionel) is described as not returning to pick up his new shirt in November 1958, and possessing love letters dated from 1957 through early winter (November) 1958. Booth reports the fallout shelter where Lionel's body was discovered was sealed in 1958. Later Lionel is described as reported missing by his boss in January 1960, and the illegitimate child he fathered was born in 1960 as well. That would mean his boss didn't notice he was gone for over a year, and his pregnant girlfriend was with child for at least 14 months.
Continuity mistake: It is the evidentiary hearing, all characters are put on the witness stand so you get to see what all of them are wearing. The trial starts the next day and lasts at least three days, but when they are in court at the end of the trial for the reading of the verdict for the gravedigger they all have the exact same outfits on as the day of the evidentiary hearing.
Factual error: In the opening shot of the episode they show a plane landing at an airport, and it says Dulles International Airport on the screen. There is even a nice shot of the US Capitol building in the background, the problem is that Dulles Airport is about 25 miles from Washington, if the shot is real it was filmed at Reagan National Airport. (00:00:10)
Continuity mistake: During the scene when Booth and Bones are driving to tell Angela that her boyfriend is dead, the scenery behind Booth jumps suddenly mid shot. Initially the scene is of a wide plain, then suddenly a mountain range appears. This scene was obviously shot on a sound stage, with the scenic background being projected onto screens around the vehicle, an old trick in movie making. It seems that somebody mistimed the loop of the tape they were using. (00:12:30)
Continuity mistake: When Angela asks to see the girl's drawing, in the shot from the girl's perspective as she hands the pad over, the painting's quite small in the centre of the page. When it then cuts to Angela's point of view it takes up nearly the whole page. (00:02:45)
Other mistake: Hodges and Angie find a piece of paper buried with Bones' mother and they tell Bones that the date of the movie is September 22, 1993. The ticket has two dates, September 22, 1993 and in two different places it says April 22, 1993. The Fugitive was released August 6th, 1993.
Plot hole: Angela comes into the autopsy room to inform Hodgins and Cam that Brennan is awake. Cam says that Brennan said the entry wound was cold and Hodgins comes up with the ice bullet theory. But how did Cam know what Brennan said about the cold if she didn't already know Brennan was awake?
Continuity mistake: When Bones goes to open the vent on the wall in the mortuary while standing on the coffin, the shot from inside the grate shows her taking a step up and being exactly at head height for it. Only problem is that in the shot previous to this she was significantly short of the grate, and there was nothing on the coffin lid for her to step up onto.
Factual error: Temperance Brennan is examining the bones of the victim and mentions she wants to have a baby. She starts examining the tibia, mentions the she wants to have a baby, then says "capitulum" while still looking at the tibia. In actuality, the capitulum is on the distal epiphysis of the humerus, not the tibia. (00:06:25)