It looks like the answer is at least 12 million if you have a beefy server, your server software is optimized for it, you have enough clients. If you test from one client to one server, the number of port numbers on the client will be one of the obvious resource limits (Each TCP connection is defined by the unique combination of IP and port number at the source and destination).
(You need to run multiple clients as otherwise you hit the 64K limit on port numbers first)
When it comes down to it, this is a classic example of the witticism that "the difference between theory and practise is much larger in practise than in theory" - in practise achieving the higher numbers seems to be a cycle of a. propose specific configuration/architecture/code changes, b. test it till you hit a limit, c. Have I finished? If not then d. work out what was the limiting factor, e. go back to step a (rinse and repeat).
Here is an example with 2 million TCP connections onto a beefy box (128GB RAM and 40 cores) running Phoenix http://www.phoenixframework.org/blog/the-road-to-2-million-websocket-connections - they ended up needing 50 or so reasonably significant servers just to provide the client load (their initial smaller clients maxed out to early, eg "maxed our 4core/15gb box @ 450k clients").
Here is another reference for go this time at 10 million: http://goroutines.com/10m.
This appears to be java based and 12 million connections: https://mrotaru.wordpress.com/2013/06/20/12-million-concurrent-connections-with-migratorydata-websocket-server/