I worked for Social Security and encountered several people who had established new identities and then lived under them for years. One person did it to dodge a bad driving record, the others did it to avoid alimony or child support. None appeared to have any criminal motive.
Prior to about 1980, with a little knowledge and planning, setting up a new identity with a "good" Social Security Number was not that difficult. Most people who did it just lived out their life, and death under the new identity. There is little doubt many criminals did this as well, but fingerprints would trip them up if they were arrested under their new identities.
Since 1980 or so, it is extremely difficult to pull this of but no doubt some still find ways to do it.