6.11. PXE installations work fine for a while, but eventually clients no longer boot the autoinstall kernel.

Are you using xinetd? xinetd 2.1.8.9pre11 (and assumably earlier versions) had a race condition which causes the tftp server to suddenly stop responding. this has supposedly been fixed in version 2.1.8.9pre13. Also, xinetd and inetd have default limits on how many connections can be spawned in a 60 second period. See the inetd or xinetd manpage for details on increasing this limit.