Yes. SSNs are distributed by year in blocks that are granted to hospitals. If you know a person's year and place of birth you can brute force them. For example if you wanted to generate plausible identities you could just use a common Jewish last name and get the SSN block from a major hospital in NYC for a high birth year. Say, 1955. The odds that you will be able to guess the SSN for Abraham Goldstein born in Manhattan in 1955 are going to be pretty good, especially if you have some oracle that will let you guess several times.

10,000 is probably going to be the largest number of guesses needed, and if you have prior knowledge, like a distributed set of ssns from the same year and location, you can reduce the practical effective number of guesses to a few dozen.

The freely available databases of pii in the wild can be used to infer anything missing from releases like this, and that stuff can be used to inform probabilistic password guesses, and so on. It's only a matter of time before deep learning models make most common password based security measures completely transparent and obsolete.