When you open ArcMap or ArcCatalog the ArcGIS Connection Properties tool begins running in the notifications area of your taskbar. You can also find it here:
C:\Program Files (x86)\Common Files\ArcGIS\bin\ArcGISConnection.exe
Have you tried running the Connection Test through this utility (right-click taskbar icon > Run Connection Test)? Maybe it will provide more information as to why your machine cannot connect.