When you encounter this error message, it's usually indicative of one of three problems that are preventing Mendeley Desktop from contacting the Mendeley Web servers.
This may be caused by a temporary issue either on our servers, or your ISP with routing to our servers. As such, can you first check that you can access http://www.mendeley.com/ within your web browser to check if this is the case.
If this appears fine, you'll want to check that you have nothing blocking Mendeley Desktop explicitly from connecting out. This may be due to firewalls, or anti-virus software on your own computer, or further down in your network setup. As such, you can try either disabling temporarily to see if this helps, or add an exception that explicitly allows Mendeley Desktop connectivity.
Finally, you may require proxy settings for Mendeley Desktop to connect. These settings vary from location to location and you'll need to consult whomever maintains your network for this information. Once acquired, you can either set this up (or disable it if not needed) within Mendeley Desktop by going to Tools > Options > Connection.
If you could check with your administrator about your proxy settings, it may be that your problems are caused by security software on your network that is blocking Mendeley Desktop.
On Ubuntu, try installing the libqt4-webkit and libqt4-svg packages using "sudo apt-get install libqt4-webkit libqt4-svg" in a terminal. When Mendeley is run from a terminal, you should see a message indicating that it is using 'system Qt'