Have you recently seen the message The file "%@" could not be backed up after Time Machine warned you of a failure? You're not alone: many people who have installed the macOS 12 Monterey Public Beta have also received the same error. There's no fix for it; it's likely an error in how Time Machine scans for files.
This failure appears related to a recent update to the macOS 12 Monterey public beta.
This may mean your files are not properly being back up, however. I entered Time Machine on a Mac that's running this public beta and looked around through older backups even where Time Machine claims it hasn't been updated. The files all appear to be there, but I can't check comprehensively.
If you're concerned about losing data during this test period before Apple squashes this bug in a future beta or production release of Monterey, make sure all your document files are stored on a syncing service, like iCloud Drive or Dropbox, or syncing through iCloud services. It might be a good time to make nightly clones of your computer with Carbon Copy Cloner and enable in Carbon Copy Cloner > Preferences > Updates the option Inform me of beta releases.
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