To reproduce:
1. Use a computer where you have passwords to both root and a nonroot account.
2. Log into the nonroot account only (without switch-user mode).
3. Right-click Clock in its panel.
4. In the context menu, select either Adjust Date & Time or Preferences > General tab > Time Settings button.
5. Set a time.
6. When it requests the password, enter your root password and select For this session only.
7. The dialog shakes left-right.
8. Two more password attempts and two more shake responses result in the password dialog disappearing.
9. Repeat: right-click Clock in its panel.
10. Repeat: in the context menu, select either Adjust Date & Time or Preferences > General tab > Time Settings button.
11. Repeat: set a time.
12. But, this time, when it requests the password, enter your nonroot user account password and select For this session only.

By comparison, before success with the wrong password, thus before clock has accepted any password:
0. If clock has accepted your user password from above, get rid of it (change computer, reboot, or something and confirm inability to edit time via panel because lacking password).
1. Log into the nonroot account only (without switch-user mode).
2. Select Main Menu > System > Administration > Date & Time.
3. When requested, enter your root password.
4. Edit time.
5. Verify success in the panel.

Requiring a password for an account the user is already in, especially on a local machine, seems pointless, especially if the reference time (I assume there's a reference time to which increments are added for display) is stored in a root-accessible file for all local accounts (all accounts show the same time, as I found when I tested with one time change and viewed the result in 2 nonroot accounts in Gnome in Clock using Switch User).

I'm running Clock 2.24.1 in Gnome 2.24.1 in Fedora 10 Linux on a Dell Latitude C840 laptop. I did not test changing the date.

See also Gnome Bug 559538 (http://bugzilla.gnome.org/show_bug.cgi?id=559538) & Red Hat's Bug 447225 (https://bugzilla.redhat.com/show_bug.cgi?id=447225). Neither of those found which papssword works, but the latter offers a workaround.

The documentation is okay. Clock > right-click > context menu > Help command (or Preferences > Help button) > Usage link > section 2.3 > step 2 ("Type the root password").