Thanks all for your hints and tips. Here is some information that has been
requested by people:

For Jeremy, the output of ifconfig -a:

le0: flags=863<UP,BROADCAST,NOTRAILERS,RUNNING,SIMPLEX>
	inet 205.142.124.90 netmask 0xffffff00 broadcast 205.142.124.255
lo0: flags=8009<UP,LOOPBACK,MULTICAST>
	inet 127.0.0.1 netmask 0xff000000 
sl0: flags=c010<POINTOPOINT,LINK2,MULTICAST>
sl1: flags=c010<POINTOPOINT,LINK2,MULTICAST>
ppp0: flags=8010<POINTOPOINT,MULTICAST>
ppp1: flags=8010<POINTOPOINT,MULTICAST>
tun0: flags=10<POINTOPOINT>
tun1: flags=10<POINTOPOINT>

Also, the output from netstat -an -f inet:

Active Internet connections (including servers)
Proto Recv-Q Send-Q  Local Address          Foreign Address        (state)
tcp        0      0  205.142.124.90.1023    205.142.125.113.513    ESTABLISHED
tcp        0      0  205.142.124.90.6000    205.142.125.113.1063   ESTABLISHED
tcp        0      0  *.6000                 *.*                    LISTEN
tcp        0      0  *.37                   *.*                    LISTEN
tcp        0      0  *.13                   *.*                    LISTEN
tcp        0      0  *.19                   *.*                    LISTEN
tcp        0      0  *.9                    *.*                    LISTEN
tcp        0      0  *.7                    *.*                    LISTEN
tcp        0      0  *.79                   *.*                    LISTEN
tcp        0      0  *.513                  *.*                    LISTEN
tcp        0      0  *.514                  *.*                    LISTEN
tcp        0      0  *.23                   *.*                    LISTEN
tcp        0      0  *.21                   *.*                    LISTEN
tcp        0      0  *.515                  *.*                    LISTEN
tcp        0      0  *.111                  *.*                    LISTEN
udp        0      0  *.1086                 *.*                   
udp        0      0  *.1031                 *.*                   
udp        0      0  *.1030                 *.*                   
udp        0      0  *.1029                 *.*                   
udp        0      0  *.37                   *.*                   
udp        0      0  *.13                   *.*                   
udp        0      0  *.19                   *.*                   
udp        0      0  *.9                    *.*                   
udp        0      0  *.7                    *.*                   
udp        0      0  *.518                  *.*                   
udp        0      0  *.512                  *.*                   
udp        0      0  *.520                  *.*                   
udp        0      0  *.525                  *.*                   
udp        0      0  *.514                  *.*                   
udp        0      0  *.111                  *.*                   

For Mark:

I can telnet to anyone on my local network, including Linux, SunOS, Solaris,
BSDi. I can ping nunic.nu.edu, but get lots of DUP replies from hosts all over
my local net (!). I cannot ftp into nunic.nu.edu. I can also rlogin and rsh
to other hosts here, but not out. Being root does not help. If I ping local
hosts, I get no DUP's.

It gets stranger all the time.

Thanks a million guys.

-- 
Jeremy Worrells                 Undergraduate - Computer Science
jworrell@nunic.nu.edu           National University 
jworrell@priacc.com             http://nunic.nu.edu/~jworrell