Time Machine says not enough space but there is plenty of space
I've run into this before when using networked backup, also said something like 497GiB were free when actually 4TiB were free. Seems this is caused by TM not creating a large enough sparsebundle for the initial backup. I found a workaround that at least worked for me (El Capitan) and submitted a bug report to Apple.

Workaround: You should notice that the sparsebundle on the OS X server has only ~499GiB as its capacity. Manually enlarge it by running this command on the server: sudo hdiutil resize -size [number of gigabytes]g [path to sparsebundle], e.g. sudo hdiutil resize -size 10000g /Volumes/Backups/mybackup.sparsebundle. Make it as large as you need. Retry your backup.

Edit: You may also have to resize the sparsebundle's partition in Disk Utility. I didn't have to do this, but maybe that's because my backup wasn't encrypted. For instructions, see the first (not accepted) answer here: Increase size of sparsebundle for TimeMachine