Continuity mistake: When Tooms is in the cell right at the end, we see when his food is being brought over to him on the tray that there is a drink and a bun, when the shot changes to the other side of the door the drink and bun have swapped sides. When the shot changes back to the other side of the door, the tray is now the other way around and then when the shot goes back to the other side the whole tray has now disappeared.
Continuity mistake: When Mulder experiences his near-death trance state his body is lying, apparently lifeless, in a star spangled sky. As the camera tracks in on him the toes on his right foot twitch, just after Deep Throat begins to speak. The foliage covering his chest also changes position between shots. In the wide shot he is covered up to the armpits. But the first close up shows his chest is exposed. Then in later close ups it is covered again.
Plot hole: Scully's statement that she was 23 when she wrote her senior thesis in college throws off her timeline of college, med school, residency, FBI Academy, Quantico teaching assignment for two years, and X-Files assignment at 28. Her undergraduate degree in Physics should have been completed when she was 20 or 21 to allow at least 4 or 5 years for her medical training.
Audio problem: When Jerry uses the elevator, the voice of the computer reads out the floors as "twenty-third floor, fourth floor, fifth floor" etc, missing out the "twenty-" prefix until the 29th floor. The DVD subtitles list these numbers with their prefix, so this was not intentional.
Continuity mistake: In the scene where the Riley Animal Hospital sign lights up, the time comes across the bottom of the screen saying 6:21 p.m. The scene then cuts to inside the animal hospital. However, the clock inside does not read 6:21. It reads somthing roughly before 6.
Revealing mistake: When the doctor examine Ronnie's "corpse" in the morgue, he pulls out the stake. It is quite obvious that he's struggling with a wooden stake that is placed on the actor's side, not through his chest. Moreover, the camera pans up and you can briefly see Ronnie's unharmed chest.
Continuity mistake: When Skinner walks into the cafe (the time he gets shot) another man is just walking out, leaving a table with a used glass and coffee cup. The waitress is in the back, getting Skinner a cup of coffee (she couldn't have cleaned the other table yet), but when we look over Skinner's shoulder you can see that the other table is cleared and has a clean coffee cup (turned upside down) on it.