Plot hole: The Doctor leaves Martha a recording with instructions for her for when he's human. However, the flashbacks to the incident that led to him using the Chameleon Arch don't leave enough - or any - time for him to have made the recording, presenting his decision to use the device as a spur-of-the-moment one that he immediately acts upon.
Continuity mistake: When the Doctor says, in response to Martha asking how the Family can follow them, "Stolen technology. They've got a Time Agent's vortex manipulator," in a close-up of his face he reaches with his left hand for a control on the console, but in a following shot of his hands, they're right next to each other. (00:00:25)
Continuity mistake: When Martha and Jenny are washing the floor, just as Hutchinson and Baines come along and tell them off for laughing, one of the lace ribbons on Martha's cap moves between hanging in the back in shots facing her and draped over her shoulder in shots facing the schoolboys. (00:04:15)
Continuity mistake: When Martha initially bursts into John Smith's study after hearing that he fell down the stairs, the door is mostly closed. When she runs back to the door to knock on it after Joan Redfern tells her off for bursting into a master's study unannounced, the door is open wider. (00:06:35)