Time Machine Could not complete the backup
I've had this issue before and when it occurred, the system logs of the affected Mac seemed to indicate that there were filesystem journal errors on the Time Machine drive.

My Time Machine drive is mounted remotely via an Airport Extreme to simulate a Time Capsule, and it is used by several Macs. All the other Macs were still backing up fine, so rather than connecting the drive directly to a Mac and doing a Disk Utility diagnosis and repair, I just mounted the Time Capsule drive from another Mac, deleted the backup sparse bundle from it and then removed the disk from the Time Machine preferences on the affected Mac.

I then added the drive back again and it all worked.