Please do the following. Select the Desktop icon for your drive, then press Command-I to open the Get Into window. In the topmost panel you will find the drive information listed. It will show the drive's Capacity, Used, and Available space. Please take a snapshot of the window and post it in your reply. This will give me a clear picture of the drive's space usage.
It's possible that you need more storage than you think for the backup, if the backup utility saves a copy of the existing backup until the new backup is completed as a safety precaution. I'm only guessing about this because I have never seen how the backups are handled. The OS is usually pretty careful about protecting data. The OS installers do this.
How to take a screenshot on your Mac
Capture, Save or Record Screenshots in Mac OS X - Help Desk Geek
OS X Screen Capture Shortcuts
(⌘⬆︎3) Command-Shift-3 Capture the screen to a file
(⌘⬆︎⌃3) Command-Shift-Control-3 Capture the screen to the Clipboard
(⌘⬆︎4) Command-Shift-4 Capture a selection to a file
(⌘⬆︎⌃4) Command-Shift-Control-4 Capture a selection to the Clipboard
(⌘⬆︎5) Command-Shift-5 Invoke new screen grabber