Trouble establishing a server connection between two networks

I am on a network that has a cytrix connection to the company that my job supports. We are able to connect to a windows terminal server using our cytrix agent, and we are able to connect to the lotus notes domino servers with just the dns names added to the hosts file (as far as I know). We cannot, however, ping those servers. I’m just a grunt trying to lower my dependancy on the terminal server by establishing a connection to the ticket application server, but I don’t know what needs to be done. I cannot see why I’m able to connect to the mail servers and the terminal server without being able to ping them, but not the ticket application server. Most of the servers are on the same IP block, and Remote Administrator works on some pcs on the network (regardless of IP block).

Anyone know what needs to be done for me to connect to this server?