When trying to connect to VMware vCenter through the vSphere Client, you can get the following error:
Could Not Connect
vSphere Client could not connect to <<vCenter>>.
An unknown connection error occurred. (The request failed because of a connection failure. (Unable to connect to the remote server))
This can be because either the vCenter server is unavailable (remotely connect to it using RDP and check that the service is running and the server is healthy) or because the SQL server used to host the vCenter database has problems. Check that you can query the vCenter database and that the TempDB database on the SQL server is operational.
Personally, I got the An unknown connection error occurred message because the TempDB’s transaction logs were full. After restarting the SQL service and services on the vCenter server, I was able to connect using the vSphere Client.
Addendum: VMware say that you can also get the unknown connection error if IE is set to use a proxy. To fix this, in IE go to Tools -> Internet Options -> Connections -> LAN Settings and tick Bypass proxy server for local addresses