His HP Wiki article says it was "early 20th century", with no clear citation.
Is there any information clarifying that in any canon?
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The HP Wiki is mostly correct: Dippet started in the first half of the century.
I’m not sure if this is the thought process conducted by the editor of his HP Wiki entry, but we can piece it together entirely from canon statements.
Consider the following facts:
Phineas Nigellus, a previous headmaster, died in 1925.
We have his years of life from JK Rowling’s illustration of the Black family tree; he died at 78. That is easily young enough to still be headmaster upon death.
You only get a portrait in the head's office if you die in the position.
Instead, consider this question from the Scholastic Open Book Tour in 2007 (emphasis mine):
Is Severus Snape’s portrait in the headmaster’s office?
Some have been asking why hasn’t the portrait appeared immediately. It doesn’t. The reason is that the perception in the castle itself and everyone who was in the castle, because Snape kept his secret so well was that he abandoned his post. So all the portraits you see in the headmaster’s study are all headmasters and mistresses who died, it’s like British royals. You only get good press if you die in office. Abdication is not acceptable, particularly if you marry and American.
Dippet was in office by 1942 at the latest.
We know he was headmaster when Tom Riddle opened the Chamber of Secrets for the first time, which took place in the 1942–43 school year.
Since Nigellus has a portrait in the headmaster's office, he must have died in office. Thus Dippet must have started between 1925 and 1942. Whether you call that “early” twentieth century is somewhat subjective, but it’s definitely in the ballpark.
It’s possible to put tighter bounds on Dippet’s start date, although I don’t know if we get into “early” territory yet. Consider further:
Dumbledore had been there for a few years by 1938.
This is the year in which he goes to pick up Tom Riddle from his orphanage. This is a fairly important task (I think it’s the responsibility of the Deputy Head, but I’m not entirely sure), so it’s not something he’d be doing as a new starter.
Dippet was headmaster shortly after Dumbledore joined.
In commentary in The Tales of Beedle the Bard, Dumbledore tells of a pantomime incident when he was “a young Transfiguration teacher”. The aftermath involved Dippet banning all future pantomimes.
I don’t think we know if Dumbledore was recruited under Dippet. But this means Dippet must have been in place by the late 1930s, so we can narrow his start dates to about a ten-year period.