So servers should die with their admins?
Ha, ha! I like it!
But seriously, the normal course of action is to terminate sysadmins who are incapable of producing the proper credentials to the equipment they manage. "Termination" doesn't mean "killing", just "firing from employment 'for cause'".
There was an really interesting case of a network sysadmin for San Francisco municipial government that went insane (schizophrenia/paranoia) and refused to disclose passwords to the Cisco equipment which he was supervising. Sorry, I don't have a link handy.
Hehehe, but I meant it literally too...
What if an admin dies? Should access to the servers die with him?
I would argue: no. So - in addition to your "mental illness" example - there are definitely reasons why several people should always know the root passwords to any servers (except of those which are so personal they actually should "die with their admins"... obviously...).