Does Time Machine Copy Renamed Files?

Solution 1:

Anytime you make a change to a file Time Machine makes a new copy of the file:

John Siracusa's Mac OS X 10.5 Leopard: the Ars Technica review/Time Machine:

It's unfortunate that Time Machine's cleverness does not extend just a bit further. The smallest unit of data that Time Machine will backup is an individual file. That means that if you change one byte in a 10GB file, that entire 10GB file needs to be copied to the backup volume. Hard link can't help you here. There's no way to make a hard link to "9.99999GB of the old data, plus this one byte of new data."

The beauty of Time Machine is that it automatically deletes the older copies of files from the backup when it needs to recover the space.