Years ago, births were written in family Bibles, on notes of paper, etc. when the government required birth certificates, then people of all ages applied for delayed birth certificates.
A delayed birth certificate is one that is not filed within a set period of time, in North Carolina this is within a year of the child’s birth.
If you are having trouble finding a birth certificate, check out the links below to see if your great or great-greatgrandparents filed for a delayed certificate
Delayed birth certificates for McDowell County in the USGenWeb Archives: