When Tristan shows Rory the party invitation he received from Emily, we can see that Rory was born on October 8.
This isn’t possible according to the script – Rory started school late, and she would only have been at Chilton for a short time by October 8, less than two weeks at the absolute maximum. Several weeks have now gone by, where Rory has attended multiple Friday Night Dinners, learned to play golf, helped out at a wedding, got her first D, failed to turn up in time for a big test, attended a cat’s wake, discovered her mother and English Literature teacher made a date, and confessed to Dean that she is interested in him. Furthermore, October 8 was a Sunday, not a Friday, in 2000.
If you carefully follow Rory’s adventures since starting at Chilton, you can see that her birthday would actually be October 27. A birthday in late October is confirmed in future seasons.
If you are in a nit-picky mood, feel free to join me in tut-tutting that the party invitation said that Rory was born in the morning twice in the same sentence, but neglected to mention that it’s her sixteenth birthday – no way would Emily have done such a shoddy job.
One might also wonder how Emily was able to get the names and addresses of all Rory’s classmates to send them invitations – did Headmaster Charleston let her access the school database just for her granddaughter’s birthday party?