Like the responses above have said, there is no such thing as sending fake bitcoin. However, there is a process called double-spending that is similar.

Double-spending is when one successfully spends money more than once. Bitcoin verifies each transaction added to the blockchain to make sure that the transaction has not already been spent. However, double-spending can happen, which is why most merchants that accept bitcoin payments wait for a few confirmations before you can use your bitcoins on their site.

If you want to learn more about double spending, there is an article here that explains it.