He might not end up with, as you say, a criminal record (I'll let Lawdog explain for sure, but I imagine he was just being less than clear in his terminology), but he certainly will end up with arrest warrants.
Friend of mine named Jim went to buy a gun back when the Brady Bill first went into effect. Imagien his surprise when the clerk came out of the back and informed Jim that he was denied purchase of the gun because of Jim's criminal record.
Massive bloody surprise there, I'm here to tell you. Anyhoo, once the dust settled, it turned out that a classmate of ours had been arrested and convicted on multiple burglary counts, couple of drug charges, felonies out the yin-yang.
The kicker here? Everytime he was arrested, he gave the police Jim's name, date-of-birth and SSN.
That started a nightmare for Jim that lasted a decade or so.
Folks, for God's sake get a document shredder. They aren't that expensive. Anything you throw away that has your name on it (or any sort of personal ID info) needs to be shredded.
Criminal Identity Theft is bad now. It's going to get worse. Don't wind up caught in the machinery, okay?
http://www.privacyrights.org/fs/fs17g-CrimIdTheft.htm
LawDog