Time Machine - "not enough space to back up" despite large drive

Question

Hi,
I have a 2.7GHz Quad-Core 27" iMac from July 2011. It has a 2TB HD which I back up to an external single partition 2TB HD (WD MyBook) over Time Machine (TM). The iMac HD has about 1.4TB on it.
It was "just working" until about 2 weeks ago, when I got a message that says in essence  "not enough space to back up: only 600GB available when 1.4GB is needed". There is one succesful 1.4TB backup. I get the error message hourly as it attempts to back up then fails. Previously there were multiple time stamped (houry, daily etc) backups but when I first got the error message, there was only one left, implying that TM deleted the earlier ones in its quest for more space.
As there is a 2TB drive to back up 1.4TB, there should be loads of space as each backup should just log the changes, not the whole drive.
However, TM is acting as if it needs the full 1.4GB free for each delta backup, rather than just space for the changes. This is new behaviour as it did not need this before.
I have googled the issue extensively, checked FAQs etc and found two potential causes.
The first is that I changed the name of my computer around the time that the problem appeared. This can apparently cause TM to not recognise old backups. I therefore deleted the old one and rebuilt my backup from scratch, hoping that the new name would be used. I was able to do one backup before the same issue appeared again.
The second is that the sparsebundle needs to be resized manually due to a bug, but I can't find the sparse bundle on the external HD, even using Disk Utility (this maty be because I am being very stupid - I am a recent Mac user and do not have much experience of digging into OS X's innards)
I am now at a loss as to what to do.
I would be very grateful indeed if anyone had any thoughts that they could share.
SE.

Answer 1

Semper Eadem wrote:
I got a message that says in essence  "not enough space to back up: only 600GB available when 1.4GB is needed". There is one succesful 1.4TB backup.
It's trying to do a near-full backup for some reason, and it's already deleted all but the last remaining backup trying to make room. It was probably trying to back up about 1.2 TB, since it adds 20% to the estimated size.  That's likely your entire home folder.
The first is that I changed the name of my computer around the time that the problem appeared. 
No, that shouldn't cause it.  Apparently on some of the very first versions of Leoaprd, it did, but not since.
The second is that the sparsebundle needs to be resized manually due to a bug,
No, there isn't a sparsebundle on backups made to an internal or directly-connected HD.  Those are used only for network backups.
What else did you do around the same time you changed your computer name?  See #D4 in Time Machine - Troubleshooting for common causes.  
I'd recommend turning TM off, erasing the drive and doing a "full reset" per #A4 in that same link.  Then exclude most of the large folders in your home folder, per #10 in  Time Machine - Frequently Asked Questions. Then run a backup, and immediately run another. 
If the second backup is more than a GB, use one of the apps in #A2 of Troubleshooting to see exactly what got backed-up the second time.  That may tell us what's going on.
If it wasn't large, start removing the exclusions, one at a time.  Run a backup (that should back up the excluded item(s), then run another and see how large it is.   That should pinpoint the problem.