Your default mask should be fine, don't mess with it. This is most likely caused by your firewall not being setup, most of the firewalls (i.e. Norton and McAffee) will have a setting for private/trusted network. Just add the IP address of your other computer into the list. It does not matter if you are on WiFi or on a line connection, you are going through the same router (my logical guess as you are using a linksys router).
As for workplace environment, at my company you simply have to access the server through VPN when you use WiFi, however the IP range is also different, so the WiFi network is on a separate IP range than the wall plugs.
First of all...just check ur subnet mask....it must be 255.255.255 or 255.255
& the other thing is turn off ur firewall....it restricts pinging.....& at the last it depends upon which type of lan connection u r accessing....i mean cable net? if administrator is somebody else..den he has put some restrictions on accessing 2 PC's !!!