Hello SurferTim:
Looks like it dies on opening a new connection. I modified the client.connected while loop to print a message on failed connection as that was the only place where it goes into an endless (halted) loop without flashing an LED and sure enough, that was the location where it was dying.
After some additional testing, it appears that the ethershield IS flooding the server. During a locked/poll session, the device should be phoning home every few seconds to see if it needs to change state. I did some digging around and found that just before the Arduino halted, I had over 200 incoming HTTP connections from the Arduino's IP. It appears that the Wiznet can only support so many connections and when too many connections are open, it will cause client.connect to fail. 
Every 1.0s: netstat -an | grep tcp | grep 80 Thu Aug 30 16:06:39 2012
tcp6 0 0 :::80 :::* LISTEN
tcp6 0 0 172.16.222.100:80 172.16.222.160:1027 TIME_WAIT
tcp6 0 0 172.16.222.100:80 172.16.222.160:1058 TIME_WAIT
tcp6 0 0 172.16.222.100:80 172.16.222.160:1038 TIME_WAIT
tcp6 0 0 172.16.222.100:80 172.16.222.160:1045 TIME_WAIT
tcp6 0 0 172.16.222.100:80 172.16.222.160:1043 TIME_WAIT
tcp6 0 0 172.16.222.100:80 172.16.222.160:1066 TIME_WAIT
tcp6 0 0 172.16.222.100:80 172.16.222.160:1044 TIME_WAIT
tcp6 0 0 172.16.222.100:80 172.16.222.160:1061 TIME_WAIT
tcp6 0 0 172.16.222.100:80 172.16.222.160:1056 TIME_WAIT
tcp6 0 0 172.16.222.100:80 172.16.222.160:1057 TIME_WAIT
tcp6 0 0 172.16.222.100:80 172.16.222.160:1051 TIME_WAIT
tcp6 0 0 172.16.222.100:80 172.16.222.160:1048 TIME_WAIT
tcp6 0 0 172.16.222.100:80 172.16.222.160:1062 TIME_WAIT
tcp6 0 0 172.16.222.100:80 172.16.222.160:1042 TIME_WAIT
tcp6 0 0 172.16.222.100:80 172.16.222.160:1055 TIME_WAIT
tcp6 0 0 172.16.222.100:80 172.16.222.160:1039 TIME_WAIT
tcp6 0 0 172.16.222.100:80 172.16.222.160:1060 TIME_WAIT
tcp6 0 0 172.16.222.100:80 172.16.222.160:1053 TIME_WAIT
tcp6 0 0 172.16.222.100:80 172.16.222.160:1036 TIME_WAIT
tcp6 0 0 172.16.222.100:80 172.16.222.160:1029 TIME_WAIT
tcp6 0 0 172.16.222.100:80 172.16.222.160:1041 TIME_WAIT
tcp6 0 0 172.16.222.100:80 172.16.222.160:1034 TIME_WAIT
tcp6 0 0 172.16.222.100:80 172.16.222.160:1037 TIME_WAIT
tcp6 0 0 172.16.222.100:80 172.16.222.160:1030 TIME_WAIT
tcp6 0 0 172.16.222.100:80 172.16.222.160:1067 TIME_WAIT
tcp6 0 0 172.16.222.100:80 172.16.222.160:1033 TIME_WAIT
tcp6 0 0 172.16.222.100:80 172.16.222.160:1032 TIME_WAIT
tcp6 0 0 172.16.222.100:80 172.16.222.160:1059 TIME_WAIT
tcp6 0 0 172.16.222.100:80 172.16.222.160:1065 TIME_WAIT
tcp6 0 0 172.16.222.100:80 172.16.222.160:1052 TIME_WAIT
tcp6 0 0 172.16.222.100:80 172.16.222.160:1047 TIME_WAIT
tcp6 0 0 172.16.222.100:80 172.16.222.160:1050 TIME_WAIT
tcp6 0 0 172.16.222.100:80 172.16.222.160:1064 TIME_WAIT
tcp6 0 0 172.16.222.100:80 172.16.222.160:1025 TIME_WAIT
tcp6 0 0 172.16.222.100:80 172.16.222.160:1031 TIME_WAIT
tcp6 0 0 172.16.222.100:80 172.16.222.160:1040 TIME_WAIT
tcp6 0 0 172.16.222.100:80 172.16.222.160:1054 TIME_WAIT
tcp6 0 0 172.16.222.100:80 172.16.222.160:1035 TIME_WAIT
tcp6 0 0 172.16.222.100:80 172.16.222.160:1046 TIME_WAIT
tcp6 0 0 172.16.222.100:80 172.16.222.160:1028 TIME_WAIT
tcp6 0 0 172.16.222.100:80 172.16.222.160:1063 TIME_WAIT
tcp6 0 0 172.16.222.100:80 172.16.222.160:1049 TIME_WAIT
I guess the big question now is how can I tell the Arduino to only phone home once every 30sec or so and still be able to read cards in between pollings? Because it has to be able to read a card, I think that delay(5000) is out?