I found an old thread with this title. First I tried
which did nothing. Then
sudo gconftool-2 --shutdown
sudo rm -rf ~/.gconf/apps/panel
sudo pkill gnome-panel
which allowed me to manipulate the panel in the normal way, but only while that process was running.
sudo debconf gnome-panel
The old thread was solved by creating a user. That new user could manipulate the panels, then for some reason so could the old user.
Can I get a permanent fix? Using Lucid, if it matters.
EDIT: Nevermind. Logged out and back in. It all works now.