There are dozens of reasons why a Virtual Network Computing (VNC) client will not connect properly, though most commonly it is due to the server's router refusing to let connections through on the proper ports. This can be fixed either by the VNC server's system administrator or, for one-time use, there is a workaround called "listening mode" that comes with most free third-party VNC clients. Enabling listening mode requires that you forward some ports on your own router.
Ask your VNC server's system administrator to navigate from the actual machine VNC is running on to GoToMyVNC.com and run the test to see if the machine is able to accept connections. If it is, then there is an application on your own machine, such as the Windows security application, that is blocking your VNC client. You will need to open the security application's preferences dialogue and change the settings so your VNC client can use the Internet connection unhindered. If the VNC server is unable to accept the connection, ask the system administrator to forward the proper ports or to connect to you in listening mode.
Connect to your own router's web-based configuration page by navigating to "http://192.168.0.1" in your browser. If your local IP address is not within the "192.168.0." range, you will want to change the numbers preceding the ".1" to reflect that. For example, some networks are set-up in the "192.168.1." block. Furthermore, some routers will require you to append ":8080" to the end of the IP address to access the configuration page.
Log in to the Administrator account using the proper details. If you do not know your login details you can try the default passwords for your router. These can be found on dozens of websites or from your manufacturer's website. If those do not work, try contacting your Internet Service Provider to see if they changed your router's passwords when they set-up your Internet connection.
Change the NAT/Firewall settings to allow connections on ports 5500 and 5900. Port 5500 is required for listening mode, while port 5900 is required to allow incoming connections if you are running a VNC server of your own.
Start your VNC client in listening mode and have the system administrator of the VNC server attempt to connect to you using a reverse connection---where the server queries the client for a connection first.
Many operating system's pre-installed remote desktop and VNC viewers do not support listening mode. Try a free third-party VNC software suite if your default client doesn't have the feature.