At Nearly Headless Nick's deathday party in that book, his death is stated to have been on 31 October, 1492.
Since the celebration was commemorating the five-hundredth anniversary of Nick's death, this seems to say the scene takes place on 31 October, 1992. Rowling confirmed, via her Twitter account that that was the day the Epilogue was set.
But in one scene of Harry Potter and the Goblet of Fire, Molly Weasley reminisces about Ogg, Hagrid's predecessor; since Hagrid's years as gamekeeper appear to overlap with Molly's years at school, it seems unlikely that she remembers a previous gamekeeper.
However, it is possible that Hagrid did not become a gamekeeper immediately after being expelled, but rather started as Ogg's assistant.
It is possible, however unlikely, that Fudge could be referring to a more distant predecessor, as Thatcher took office before Fudge supposedly became Minister for Magic.
(Although one could imagine a scenario in which Fudge was delegated by the Minister for Magic to meet with the Prime Minister while he, Fudge, was in an inferior office.) Additionally, the personality and history of the Prime Minister portrayed appears to more closely match that of Tony Blair than of John Major.
For example, in Harry Potter and the Goblet of Fire, both 1 September and 2 September are given as Mondays and, in Harry Potter and the Prisoner of Azkaban, Buckbeak's trial is set on 20 April, but careful parsing of the text reveals that it could have happened no later than February.
Also, in Harry Potter and the Goblet of Fire it is said that the Champions were chosen on Saturday 31 October, but 31 October was a Monday in 1994.
If the books were set in sync with publication date of Philosopher's Stone, then this event would occur in 2002 (after Blair became Prime Minister).
In Harry Potter and the Order of the Phoenix, the Dursleys are seen to have a flat-screen television, which were not commonly available in the mid-1990s.
In addition the Dursley's car, shown in the same film, bears a 2006 year registration plate.
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Fans have created a timeline of the Harry Potter series from a single piece of information in Harry Potter and the Chamber of Secrets.
A similar problem involves Nicolas Flamel, who was mentioned as being 665 at the time of the first book, but Nicolas Flamel was a real historical figure and would not have reached that age until 1996, around the time the first book was published.