The kicker is they reuse the SSNs. Had a gig where I was working with the 'death master' files. (It use to be much easier to get your hands on the files - https://ladmf.ntis.gov now) and it would list every SSN number used by the dead. Lots of duplicate SSNs, some malformed data, some just missing. An interesting data set to look at, and a great reminder why folks should not be using this as an identifier.
Surely not, there must be a day in which two children were born on the same day in the same hospital in New York City to two women named Jane Smith?