Although that case is real, it did not happen in the US; I believe it was Russia or some other former Soviet Republic. Under systems of law evolved from British common law, it is generally held as necessary to inform the other party of such a change to the contract.

Sending bills for services not rendered can actually result in payment from large corporations because they are constantly receiving bills, so if it looks right there is a chance someone will just pay it. However, I believe it is also fraud if they notice and can thus get you in trouble; remember, the law is primarily there to protect companies and rich people.