Well, 10.5 is the same as 10.4. Go to Applications/Utilities/Network Utility and click on Ping in the screen that comes up, enter your ISP's URL (http://www.comcast.com, for example), select the number of pings to send, 5 is a good number, and start the ping. It will tell you the ping times (anything under about 30 msec should be ok) and if there were any lost packets.

Another thing to try: if you can connect the G5 directly to your internet connection, bypassing the router, try that and see what happens.

If all else fails, try repairing permissions. That sometimes fixes otherwise inexplicable problems.