I actually just fixed this on the server that was locking up.
I took a look at the syslog on an off chance and saw a whole bunch of errors like:
warning: mail_queue_enter: create file maildrop/106333.1850: Permission denied
So I chmod'd the folder to correct the permissions and now the problem has gone away.
The machine I'm using is a VM clone of another machine so perhaps the permissions were screwed as part of the cloning. If can't think for the life of my though why a postfix error would stop the assignment grid from working.