Are you sure you're pinging the correct IP address. You've mentioned both 192.168.0.5 and 192.16.0.5. The first one is more 'standard'.
Also, if you're trying to browse one machine from the other using 'My Network Places', the chances are it's not working because you may have to enable NetBIOS over TCP/IP first. (It works differently if you simply try to map a network drive or access it from the command line, as described already.)
When you go to "Browse the Network" / "My Network Places", you are using the browser service. Unfortunately, the browser service uses NetBIOS over TCP/IP.
So, you need to check NetBIOS over TCP/IP is enabled. Bring up your Network Connections. Open the "Local Area Connection" that you are having trouble with... (not the one connected to the Internet). Open Properties. Double click on "Internet Protocol (TCP/IP)" in the list. Under the General tab, click Advanced. Click on the WINS tab and then check the box "Enable NetBIOS over TCP/IP". That's it.
While you're at it, check the Computer Browser service is running. Go Start -> Control Panel -> Administrative Tools -> Services. Make sure the "Computer Browser" service is started and set to Automatic.
With any luck, this will allow you to browse machines in the Workgroup and should also resolve your file sharing problems.
Also, what firewalls are you running? It might be a good idea to disable them just while you're trying to diagnose the problem. Note: I'm not advocating turning firewalls off for more than a few minutes!!