During WWII, some men who were considered too old for entrance into the Army by just a month or a few weeks were allowed to join if they provided proof that they had been born prematurely. Their “birthday” for purposes of enlistment was the day they would have been at 40 weeks gestation. I don’t think they actually got issued a new birth certificate-- I think an addendum note was simply put in their file-- or maybe it was a waiver. Given the patriotic fervor and rush to join after Pearl Harbor, I wouldn’t doubt some of the affidavits of prematurity were lies.
Dorothy Parker was born prematurely, and wanted to get into a program for overseas correspondents during the Spanish Civil War, but she was too old by months. She tried to get a waiver, as some men had, just as in WWII, based on when she would have been full-term, but was denied, probably due to sexism.
There are more people than you’d think whose birthdays are guesses. Babies given up at Safe Havens get birth certificates with birthdays that are best guesses, mainly based on the healing of the umbilicus. I suppose if it ever came down to something like they were a day too young to run in a presidential election, they could challenge the assumed age in court, and might convince a judge to make them a day older.
I’ll bet there are people who got the wrong date of birth at Ellis Island due to Europeans writing dates differently from Americans.