On a mountain halfway between Reno and Rome/We have a machine in a plexiglass dome/Which listens and looks into everyone’s home.–Dr Seuss
How many people with the same name share the last four digits of their SSN? Here is the start of a calculation:
- 300 million people in the USA
- 3 million named Smith
- 45 thousand named John Smith
- 1 million pairs of John Smiths
- 100 John Smiths with the same SSN (last four digits)
John Smith is only one common name–John Johnson, Mary Smith, William Jones …
To carry this out further, one would have to consider every name belonging to more than one person; I don’t know how to do that.
Here is another approach:
- 300 million people in the USA
- 30,000 with the same SSN (last four digits)
- 500 million pairs of people with the same SSN (last four digits) who might share a name
Evidently to check that everyone with the same name and the same last four digits in their SSN are in fact different persons would be a demanding task. Computers might identify cases of interest, but to resolve them would likely require human effort.
New NY 23rd 
Our president has made is that dead people vote. Your model does not have to include only living John Smiths, but every John Smith that has been ever had e Social Security card since it started in 1935.
Go to Facebook and see how many members have your name.
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I once knew a man who had two driver’s licences–a clean one for getting insurance and another for getting tickets, he explained. Using his licences as ID, he probably could have voted twice if he wanted, but more likely he didn’t vote even once.
National ID numbers would solve such problems, but Americans have long resisted that solution. SSNs aren’t intended to serve that purpose.
How secure is your SSN if the last four digits are known? Not as much as one might think.
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