This may not be the best place for this, but after hours of fruitless googling on how to finish or restore a partial time machine backup I just stumbled on a simple solution to a devastating problem and thought it might be of use to others...
Just before selling my old MacBook Pro I dutifully backed up using time machine, then zeroed my hard drive. I found out later the backup had failed and spent the best part of a day in deepest depression.
Fortunately the backup seemed to have failed near the end of the process, and I was able to recover my files:
* In the appropriate backup directory on your external HDD find the offending backup package, called "[year]-[month]-[day]-xxxxxx.inProgress
* Choose "Show Package Contents"
You should be able to open and save files to a safe place on your internal drive (as long as they were backed up completely before the backup failed).
Time Machine, OS X Mountain Lion (10.8.2), MBP 2.5 GHz Intel Core i5 4Gb/.5TB