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Continuity: When Harry and Hermione leave from the hospital wing using the Time Turner, Dumbledore says it is ten minutes until midnight, but on the way back to the ward after they rescue Sirius, they hear Dumbledore say to the other versions of them that it is five minutes to midnight. (Corrected in later versions.)
Factual error: Chapter sixteen (Professor Trelawney's Prediction) takes place in June 1994. In the beginning of page 341 (Bloomsbury edition) of that chapter, Harry Potter reads a note from Hagrid and says, 'Buckbeak's appeal - it's set for the sixth.' The night after the Hippogriff's appeal, Remus Lupin is forced into a werewolf form by the full moon, which means that this happened on the night of June 6th, 1994. The web page http://www.hp-lexicon.org/timelines/main/timeline_1990s.html#1994 also backs this up. However, a brief look at lunar phase calendar sites such as http://stardate.org/nightsky/moon/index.php?month=6&year=1994 and http://mysite.verizon.net/bonniehill/pages.aux/lunar/lunar.phases.1994.html definitely shows June 9th, 1994, as a new-moon night, which means that a full moon could not possibly rise three days earlier.