Can't rearrange Dock icons

On OS X Yosemite 10.10.2, I can't rearrange my dock icons. I click one, and drag, but it just stays in place and stops highlighting. Here's a demonstration (GIF).

This has been so ever since I upgraded to Yosemite, and no version of it ever fixed it.

After restart, I can drag once or twice but I'm not sure what event causes it to stop. If I wait too long after reboot, that privilege is taken away from me.

Any idea what can be done?