The alien is supposed to be held until the matter is resolved. Since a quick check with the DHS database will show whether such claim is valid, the adjudication can happen very quickly.
Furthermore, even the relatives quoted in the news reports admit that those workers were illegal aliens. I seriously doubt that most of those released denied being illegal aliens. They are told that lying to a fededal agent is a felony, and they act accordingly.
Chertoff has become a parody of himself. I hope that SNL does not get hold of this news tidbit.