My wife was born in 1946 and she had no birth certificate. She was born in her grandmother's log cabin in Northern Michigan. In fact, it was her grandmother who delivered her, just like she did my wife's sister 14 months later. It wasn't until she was nearly 30 before she finally got a birth certificate from the state of Michigan. But not having a birth certificate, at least back then, didn't really mean much. She got a Social Security card when she was 15, got into college when she was 18, a driver's license when she was 19 and was able to register to vote when she turned 21, and we were able to a get a marriage license in 1967, all without her having a birth certificate.
Now along the way she actually tried to get her birth certificate once, but she was told that the information that she gave them wasn't correct and her parent's excuse was that they'd never actually had her birth registered. Anyway she got a big shock one day when an old friend of her parents accidentally (he was not aware that she didn't know the story) mentioned something about her father and how his marriage to her mother hadn't worked out. The problem was that she had never heard of this person before, and he had a different last name than she did. She confronted her mother, who finally admitted that yes, she had been married to this other guy and when she got pregnant for my wife, he left her. Shortly after she was born, her mother met someone who later became what she had always believed to be her father. She got pregnant again, for my wife's sister, and they were married shortly before she was born, so her sister had his name, but technically my wife was born with a different last name, but she was never told this and she was never adopted by her stepfather. What that meant was that a birth certificate did exist, just that it was under a different name so that's why they never found it (this was back before everything was computerized and the people in the country clerks office was just to lazy to try matching dates instead of just names).
Of course, now we had this dilemma, my wife had been living her whole life using what was an assumed name. We consulted an attorney and it turns out that no laws had been broken and since, after we had been married, my wife, now with a marriage license as proof, had had all of her 'official' records changed, such as Social Security, voter registration, driver's license, etc. She never went back to her schools to update them, since no one had ever asked to actually see her diploma except when she applied to college but back then she was still living under that same name as was on her school records (note, she never finished college).
BTW, a few years back, we had both of our DNA's run through Ancestry.com and she actually found her father's family (he had died a few years before that). She never attempted to contact any of them because she was not sure if they knew the story and since none of them ever tried to contact her, they either missed the connection (hey you have this heretofore unknown sibling/first cousin) or they caught it but never wanted to go there.