We don't know the exact dates for when Hogwarts was founded, but can you at least give bounds or make an educated guess? Alternately, what are the bounds for the date of building the Hogwarts castle, or the dates of birth and death of the Founders.
Here are a few starting points I could think of.
Professor Binns, who teaches History of Magic, states in CS chapter 9 (this has happened in 1992)
‘You all know, of course, that Hogwarts was founded over a thousand years ago – the precise date is uncertain – by the four greatest witches and wizards of the age. The four school houses are named after them: Godric Gryffindor, Helga Hufflepuff, Rowena Ravenclaw and Salazar Slytherin. They built this castle together, far from prying Muggle eyes, for it was an age when magic was feared by common people, and witches and wizards suffered much persecution.’
In GF chapter 12, the sorting hat's song starts like this.
‘A thousand years or more ago,
When I was newly sewn,
There lived four wizards of renown,
Whose names are still well known:
The Wizard of the Month Archive page of the old J. K. Rowling site: the founders are listed as “Medieval (precise dates unknown)” even though dates are given for many other wizards on that page.
- The castle must have been built at least within a lifetime after founding the school, because Salazar Slytherin has built the Chamber of Secrets in it. Of course, wizards can have a very long lifespan.
- It would be nice if the Hogwarts coat of arms, which is displayed at the beginning of Harry Potter books, gave the year of founding. Alas, no such luck. It only shows a motto, and the motto or the coat of arms could be adapted much later than the school was founded.
- The names of the school and of its founders gives some information. It sounds unlikely to me that noblemen living in Scotland would have had names like Godric Gryffindor over a thousand years ago. Of course, this could be a later form of his name.
- Similarly, castles similar to the one Hogwarts is described were started to be used only about a thousand years ago.
- In DH chapter 31, The ghost of the Grey Lady talks about where she's hidden the diadem. She says she has hidden it in “A forest in Albania.” Would she really has used the name place “Albania” like that had she lived as early as the 12th century? Seems curious.
- Like Slytherincess notes, the Pottermore website says Merlin (yes, that Merlin) was a student of Hogwarts.
- We know how Rowling has claimed she was “bad with numbers” and has made that mistake about the Playstation. However, she's a teacher and an educated person in general, so she surely knows more about the medieval history of Britain than I do.
Were it not for Binns's statement, I'd happily think Hogwarts must have been founded between the 15th and 17th centuries. However, Binns is an authority in this and he's made that statement right in book canon.) Update: with the Sorting Hat's evidence, I retract this last paragraph.
Update: added the entry about the Sorting Hat's song in Harry's fourth year.