I assume you're using DSL? That seems to be the most common connection to have issues with. Is there a way you can manually reconnect? By right-clicking the connection icon, perhaps?
I think this happens because of the way your computer "signs on" using PPPoE. When you hibernate and come back out of it, your computer just comes up as it was, except that DSL connection has long since timed out. It may be that you have to use shut-down instead of hibernate.
i actually am using cable, not DSL. This always happened with 98 and ME too. I just never really looked into it. If I shut down completely for more than like 30 secs without rebooting, I lose the connection and the main PC has to be rebooted in order for me to connect again. I was just hoping to find a little tweak for this, but if there isn't, then so be it. I will survive.
With DHCP on, run ipconfig /all at a command prompt and record all of those settings. Then go into your TCP/IP settings and manually configure all your settings and see if that works. Just make sure you remember that if your connection is lost after awhile on all your computers, you'll have to turn DHCP back on because your ISP reassigned IPs so your static IP will no longer work.
Or don't hibernate.
:edit: waitaminit...where is this computer in your network setup??