It looks like, from an EU perspective, this would be acceptable.
http://europa.eu/abc/travel/doc/index_en.htm
When entering or leaving the EU at the external borders you will need a valid passport or an ID card.When entering or leaving the EU at the external borders you will need a valid passport or an ID card. You may, of course, need your passport when leaving the EU in order to enter the country of your destination.
As to why one would want to do this:
passports get lost and stolen.
Rather than spending time and money at a consulate, one in this situation could simply return home and get a new passport at that time.
Now, visitors to the US should make sure that their I-94 gets processed recording their departure.