This was actually not an uncommon uncertainty during the Tudor era (and in predating eras) — it was only when Thomas Cromwell issued a mandate for strict parish registers in 1538 that there was a regulated and widespread record of christenings, burials, and marriages. Considering that the mandate was implemented decades after Anne Boleyn (as well as many of Henry’s other wives, in fact) was born, the uncertainty of Anne’s date of birth is likely due to poor record-keeping in general.

There are, however, more specifically recorded dates relevant to the estimation of Anne’s exact age, such as when she attended boarding school, that can be placed in the social context of Tudor England to establish a likely timeline of her life — but even here the dates recorded are inconsistent. Frustrating, right?

There are a few excellent arguments that people have written about Anne’s age that will be more in-depth and well-researched than what I can give, so I recommend looking here at Gareth Russel’s proposal of Anne being born in 1507, and Eric Ives’ book “The Life and Death of Anne Boleyn”.

My answer is somewhat short, but I hope it helps a little!