Time Machine backup never finishes

If the erased volume still doesn't allow a complete backup try booting into the Recovery volume (boot with the Command + R keys held down), select Disk Utility and run First Aid on your boot drive. There may be a damaged file(s) on there that are interrupting the TM backup.

Note: if your problem Mac is an M1 then the recovery boot is as follows:

To start your M1 Mac into Recovery mode, shut it down. Then power on the Mac, but hold the power key until “Loading startup options” appears. Then proceed as needed.

Use macOS Recovery on a Mac with Apple silicon - Apple Support

Try booting to Safe Mode. That boot up takes minutes and clears caches and other things that may resolve your problem.

How to use safe mode on your Mac - Apple Support

Can you backup while in Safe Mode?

A lot of times just boot to Safe Mode followed by normal boot resolves problems.

LarryW-PDX wrote:

The external drive in this case is formatted using APFS, has a single Container with three Volumes, all sharing the same physical hardware. It seems implausible that two of the three machines using the drive could perform Time Machine backups without problems while the third machine fails to do so. To my understanding APFS uses an intelligent system to share sectors on the drive among the various Volumes dynamically, so the same physical sectors are used by all three Volumes. Can you suggest a failure mechanism which would selectively impact only one of these Volumes?

APFS—This should be in quotes " intelligent system"...

I would delete the Volume and add it back in —and compare your results.

I have always had better luck with a dedicated drive for TM— not 3 volumes....

this goes way back to "partitions" and how easily TM is confused.

Add, delete, or erase APFS volumes in Disk Utility on Mac

I suspect you are better off with three 1-TB drives compare to one 3-TB drive with 3 Volumes under one Container.

Drives can fail at anytime, not a matter of "if", but a matter of "when."

Post back your success or failure...

I can suggest no "failure mechanism"—TM is finicky enough as is without complicating its function.

3-2-1 Backup Strategy: three copies of your data, two different methods, and one offsite.

LarryW-PDX wrote:

Time Machine backup never finishes

I have a new Mac mini with a 4 Tb external drive attached. The drive is formatted to have three APFS partitions, one each for backing up the mini and two networked MacBook Pros. All devices have the latest Monterey software. The mini and the older (16-inch 2019) Pro back up with Time Machine just fine, but the newer (13-inch 2020) Pro hangs when backing up. TM will prepare the backup, copy some tens of Mb, then hang with a 'Calculating time remaining ...' message. I've tried rebooting all machines, stopping and restarting Time Machine, unmounting/first aid/remounting external drive, and various other voodoo smoke-and-mirrors incantations. I'm going nuts here, and could use some help.

If you see—

"Time Machine has completed a new backup for you, or get a message saying it can't create the backup,"

or “failed to complete the backup, ” or “Preparing backup”

These message reappears most likely the TimeMachine disk is operating in a state of failure and will continue to fail with increasing frequency, in this case replace the drive.

LarryW-PDX wrote:

Any suggestions how to file this bug with Apple?

Use the same link as here for a bug report and/or user feedback: http://www.apple.com/feedback 

Can you try erasing the one partition and then starting a new ™ backup?

Is the backup disk layout with one Container and three Volumes, with one Volume for each Mac?

As a test do you have a spare disk you can connect and use that disk solely for this Mac. Erase it and start a new backup.