In what cases would a bitcoin address not have a private key?
If an address merely 'looks like' an address and all other aspects of a transaction are ok then miners will include it in a block. 

This has been (ab)used to embed ASCII art into the blockchain at the cost of permanently destroying some bitcoin.

For an example with explanation see https://bitcointalk.org/index.php?topic=33618.msg420597#msg420597