Connection Timeout Expired. The timeout period elapsed while attempting to consume the pre-login handshake acknowledgement
We had a situation where our sql server, a vm, went down with the ESX host it was on.  

The vm rebooted, but TCP 1433 would not let us connect with the posted error. 

We confirmed that all TCP settings were as intended in SQL Server Configuration, but ultimately the resolution was to change the IP of the server. Weird right?

My only thought is that something at the VMWare level was causing traffic to be dropped, but had no view into that platform as it was in the 'cloud.'

I don't know if is useful, but I hope someday it helps someone else.

Other notes are that SQL started fine, nothing in error log, with agent disabled we could access sql via pipes and udp 1434.