What can I do when my Time Machine password is not accepted, although it's correct? Having troubles with a botched system upgrade on OS X 10.10.3, I tried to restore my MacBook Pro, 15" Early 2011 version, using an encrypted Time Machine backup made on an external USB drive.

The backup is in working order; I can boot from it just fine. However, when I try to restore from a previous backup version through the system restore feature, I am asked for my Time Machine backup volume, and when I click the external drive, and enter the password, there's a dialog saying the password is wrong.

But it isn't.

I am a hundred percent certain it's the correct password. It can't be copy-pasted, unfortunately, but I've typed it often enough to know it's correct. And when I finally got to boot the Mac after the – what I thought was – a botched update, I checked my Keychain to verify that the Time Machine password was actually the one I had used previously.

Has this ever happened to anyone, and is there any solution to this, apart from maybe creating an entirely new Time Machine volume and, gasp, not encrypting it?
I would delete the existing keychain entry for your backup and re-enter.  It is very possible that you have saved an incorrect value to your keychain.