If it would be your server and your customers, what would you do instead?
1) Is this a vserver that uses openvz or virtuozzo? If yes, then check with:
if you hit any limits.
2) Try to get more verbose infos from postfix: http://www.postfix.org/DEBUG_README.html
3) E.g. create a new server from scratch as virtual server (works fine on a desktop pc with vmware or other free virtualisation software) and then check if you can reproduce the problem there and if not, compare the relevant config files for postfix and mysql.