Problem: 
Unable to log into a Drupal site.  Logging in with valid credentials would fail without any visible error in the UI.

Solution:
Repair the sessions table.

When I exported the database using Sequel Pro, I got the following error:
“Table ‘./d6_eiper/sessions’ is marked as crashed and should be repaired’.  I was able to do this with Sequel Pro, but you could also use phpMyAdmin.

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Update 4/6/2012: More tables starting crashing in the same way and we were able to repair them as well.  The reason for these crashes is still a mystery.  We restarted the web server and haven’t seen any crashes since.