I have a slackware machine set up on my network acting as a fileserver that I access with my windows machine via SSH for command line stuff, and VNC when I want X windows access. For some reason it seems to lose network connectivity very often when I am using the terminal or have an X window open. I get the message "connection reset by peer" from VNC. Usually I can reconnect after a few seconds. Any ideas where to start? Specific log files to look at, etc? I used to use the machine with slack as my main PC, and never had networking issues....

There are lots of places that an idle timeout can occur. See this thread.

I wouldn't say its timing out...I lose the connection in the middle of doing something on the command line or an X window. Plus its not just ssh, it happens when I am browsing files on the network in windows as well.

Then you have flaky connectivity. Check your logs for errors on the Ethernet port(s).