I have been meaning to write this down for quite some time, but it always eluded me. When you are using Time Machine to back up your Mac, you get the chance of password-protecting your backups.
Whenever the machine is restarted you will have to unlock the disk by entering the password, unless you store the password for the backup disk in your keychain. For paranoia's sake (and to keep the password fresh in my memory, just in case) I do not store the password in the keychain.
Now, what happens quite regularly to me is this: I reboot the machine for some reason or other, and while it is doing that, I leave the room and do something else. Before I know it, maybe an hour has passed before I come back. In the meantime, the external Time Machine drive has gone to sleep, because it was not used for an extended period of time. On the screen, there is the password dialog dutifully waiting for me to unlock the protected volume.
As soon as I do so (being very sure the password I type is correct) I hear the external drive spin up again. Because I use a dual-drive RAID enclosure it takes a bit longer to get ready than most drives, because it powers up the drives with a slight delay to go easy on the power supply. All in all I estimate it takes about 10 seconds before the drive is ready. This, however, seems to be longer than the OS is willing to wait for the drive to report back, making it believe that the password was wrong. It will ask for the password again and again, and no matter how often you make sure you typed it correctly, there is no chance of convincing it to accept it.
When this first happened to me, I was getting a little panicky, because I thought that either I was losing my mind, not remembering the very same password I had been entering for months, or that something might be wrong with the drive. Turns out, everything is fine, the drive is mounted with the first attempt, it is just the dialog that remains open and drives you crazy - which, of course, you will not realize if you have a bunch of windows open, preventing you from noticing the icon on the desktop. Watch this short video for of demonstration of what I am talking about:
As you can see, the backup volume is mounted and the dialog is still up. So if you run into the same situation, don't worry. Just cancel the dialog and the backup will run normally.