We were unable to connect to ...
Error: Proxy server got bad address from remote server (verify ther server is running).
The issue may be that your trying to access an older version that isn't compatible, administrative connections can only be made with ArcGIS Server and ArcGIS Desktop products that are on the same version.
ESRI has a a technical article on an error message that may be relevant to your issue here
Just had this happen to us and my colleague figured out that there was an update on the server and a rule normally allowing traffic through port 443 was changed during the update. Changing the firewall rule back fixed the issue.
Check ArcGIS Server Manager to verify the machine (ArcGIS Server) that you can connect to. If you remote desktop connect to different machines, they may show in ArcCatalog, but you will receive that error message.
The problem we usually have is that the License Manager in ArcGIS Administrator is not correct. If License Manager is not correct, then ArcMap won't allow you to connect to a GIS.
Maybe your LAN settings in Internet Options have been changed! Check this.