Undocumented, as we all know, doesn't mean someone who's lost his papers, but an illegal, who never had legal residence in the first place.
One of the jokes in the Late Night Roundup did was this from Jay Leno:
If you left your driver's license at home, you are "undocumented:" if you're driving without a license, you're a criminal.
I remember hearing a story about a detective who was asked for spare change by a vagrant.
He showed the guy his badge, and aked him for his ID, and the vagrant said "Er, sorry, I left my wallet on my dresser at home."
The cop said "I've heard that so often, that sometimes I think there's a dresser somewhere in this town with a pile of wallets on it three feet high."
He was wrong: it's in Guadalajara.