ping machine.domainis all you need to quickly establish whether there is a network connection to that nachine at all, but see the ping manpage for more options.
sune>traceroute ukqcd2.ph.ed.ac.uk traceroute to ukqcd2.ph.ed.ac.uk (129.215.73.54), 30 hops max, 40 byte packets 1 gateway109.liv.ac.uk (138.253.109.1) 1.187 ms 0.915 ms 0.876 ms 2 mr-cs4msfc.liv.ac.uk (138.253.110.14) 0.795 ms 0.760 ms 0.737 ms 3 gatewaynnw.liv.ac.uk (138.253.119.10) 1.354 ms 1.150 ms 1.057 ms 4 194.66.24.250 (194.66.24.250) 2.945 ms 3.104 ms 2.404 ms 5 * * * 6 * * * 7 * * *In the above example the problem appears to be at the connection between Liverpool University and the outside world.
There is more information in the manual page
(man traceroute).