Some people (especially older women) don't know their own SSN, just their husbands'. If they never had a job, there wasn't much use for an SSN. And if their spouse passed away, they always had to use the deceased's SSN to collect survivor's benefits (or whatever it's called).

I've seen it more than once when working with health records: two people have the exact same SSN, but different sex. If I need to match records, I'll use SSN and birthdate, knowing even that's not immune to errors.