A lot of the fanfiction seems to like to give Hermione her letter in September to coincide with her birthday. That just doesn't seem very logical for the Statute of Secrecy, as you then have 10 out of every 12 10-year-old wizards and witches knowing about magic while finishing the school year (assuming even distribution of birthdays).
It seems to me that this is just asking at least one of them to brag about knowing magic and spill the secret.
Also this would mean that the poor kid born on 30 of August would only have 2 days to adjust to being magical before being shepherded off to a boarding school in the middle of nowhere.
From another point of view, getting such news on your birthday can screw up a lot of celebration plans.
Also, from rereading the memory from Dumbledore where he introduced magic to Tom, there is no mention of snow or even closeness to his birthday from the caretaker.
I believe this view came from the fact that Harry got his letter on his birthday but that piece of fanon seems to forget that letters were being sent up to at least a week or 2 beforehand (leading to the Dursleys hiding on a rock in the sea in the first place).