I hope for you, because if its as you claim, your neighbor, schoolmates and/or colleagues can have a credit card under your name pretty easily, let alone all of your family and friends.
As Kasz said, it can and does happen. The problem is what can you spend the money on without getting caught? Anything you buy can be traced back to you as an individual eventually so the ability to get away with the crime is extremely limited.
The credi card will be under your name while it isnt, its not as easy to fix as having a credit card stolen because the user appear as being legit, you will not know until you ask for a credit report. Its a good thing that its not that easy to do this here.