I thought I'd provide more info... Here's the lowdown on my problem.
Today when I logged in as admin, I noticed a green flash message that informed me about successful update of Plesk Panel. I couldn't figure out what that update was, because the system overview was still showing the same patch level as I already saw a few days ago: 11.5.30 Update #31. Here is the related info from the action log:
127.0.0.1 admin [2014-01-26 03:27:15] 'New patch available' ('Update version': '29' => '30')
127.0.0.1 admin [2014-01-31 03:35:24] 'New patch installed' ('Update version': '30' => '31')
127.0.0.1 admin [2014-02-02 03:28:16] 'New patch available' ('Update version': '30' => '31')
Today is 2014-02-06. There's no log entry about patch 32 being installed.
So today I created a new webspace and discovered that I couldn't log in as that user to the Plesk Panel. I can log in as that same user via FTP and SSH.
What I tried -- all to no avail:
1. Updated user password, to rule out typos.
2. Disabled and enabling the user.
3. Changed his role from Webmaster to Administrator.
4. Created another user and another webspace.
Then I found a recipe how to reinstall microupdates, and followed it:
# rm .autoinstaller/microupdates.xml
# /usr/local/psa/admin/bin/autoinstaller --install-component base --select-product-id plesk --select-release-current --reinstall-patch
Now the system overview is showing the right version of the Panel (11.5.30 Update #32), but the problem with those new users still persists.