We do not have a precise date for the creation of the High Hay, but III 2340 marked the founding of Buckland, so it cannot have existed before that date. In practice, the Bucklanders seem to have lived for some time without any protection on their eastern border before planting the Hedge.
We have two hints about the age of the High Hay: The Fellowship of the Ring I 5 (A Conspiracy Unmasked) states that it was planted 'many generations ago', and in the following chapter Merry Brandybuck describes events involving the Hedge as happening 'long ago'. These are hardly precise dates, but they point to the High Hay being at least a century old or so at the end of the Third Age, and possibly rather older still.
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