Thank you both for your responses. This is where I'm at. I am probably dealing with multiple issues.
1) I verified that all computers used the same workgroup
2) Because one computer would give a timeout message when clicking on that workgroup (did not have authority) - I changed the workgroup name on all workgroups to a new name. My laptop no longer issued that message.
3) The pc in question that could not be pinged was previously connected wirelessly to a WAP which is connected to the router. I disconnected the wireless connection and connected via ethernet. However, then I could no longer even ping by IP address -even though the pc could get to the network.
4) The wired connection went from PC to Vonage device to router. (The vonage device has a passthrough port). I changed the connection so it went directly to the router. Name resolution now works!
So - I now know that if I want the computer accessible to the home network, I cannot connect through the port on the Vonage device. However, I still do not know why I cannot connect this PC wirelessly.
Appreciate any thoughts....