I recently had a case where Windows Backup would fail at approx. 75% complete during a full backup. The backup utility and command line both reported that "The system cannot find the file specified". The Event Viewer/Application... Services/ Microsoft/ Windows/ Backup/ Operational reads "Backup target is running low on free space. Future backups to this target may fail for want of enough space." and then at the same time and immediately after that we get "Backup started at 'TimeZ' failed with following error code '2147942402'" which means file not found or unknown error.
After a series of email communications with the Windows Backup team at Microsoft India (where, incidentally, the program was developed) the answer came back that I should run chkdsk /r and reboot the server. As this process can take hours this occurred out of hours and actually in my case needed to be repeated twice. A normal chkdsk command, run whilst the server was online, reported that the disk had errors and could not continue.
After running chkdsk /r twice, from an elevated command prompt, the backup started to work again.