Couldn't modify partition map because file system verification failed

You try to resize a volume or add a partition and Disk Utility stops with Partition failed. Couldn't modify partition map because file system verification failed. Before it changes the layout, Disk Utility checks every volume on the disk. One of them did not pass, so it refused, which is the right thing to do.

Step 1: find which volume failed

Run First Aid on each volume, then on the container, then on the disk itself. Use View, Show All Devices so all three levels are visible. The one that reports errors is the one blocking you.

diskutil list
diskutil verifyVolume disk3s1      # each volume in turn
diskutil verifyDisk disk3

Step 2: repair it from the right place

• External drive: repair from your normal session. Close any app using it first.
• Your startup disk: a mounted startup volume cannot be fully repaired. Start up in macOS Recovery and run First Aid there, from the Data volume up.
• If First Aid itself fails, our guide to Disk Utility First Aid failing goes through each message.

Step 3: clear what blocks a resize

Even a healthy APFS container may refuse to shrink while local Time Machine snapshots hold space inside it. List them, and delete them if you do not need them. Your Time Machine drive keeps its own backups regardless.

tmutil listlocalsnapshots /
sudo tmutil deletelocalsnapshots /     # all local snapshots on the startup disk

Our purgeable space guide explains what those snapshots hold.

When the partition button is greyed out

If Partition cannot be clicked at all, you have selected a volume rather than the device, or the disk is formatted with a scheme that does not allow partitions. Select the drive's top line in the sidebar.

If nothing passes

A disk that fails verification repeatedly is telling you something. Back up what you can, then erase the whole device and restore. On an external drive this takes minutes. On a startup disk, it is a reinstall.