I think the implied missing detail is that there was, or was perceived to be, a requirement to register a birth in a timely manner - currently in the UK there

I think the implied missing detail is that there was, or was perceived to be, a requirement to register a birth in a timely manner - currently in the UK there is a requirement to register a birth is within 42 days (6 weeks), but I have in my head that at some point it was two weeks. Therefore people might be tempted to lie, saying that the babe was born within the registration deadline. Or it was just that in the time between the birth happening and being registered they forgot the exact date, and either rounded it themselves, or it was done by the official registering the birth due to imprecise data ("When was the baby born?" "Two and a half weeks ago").