Hi Jerry,
I concur, no signs of other errors in the errorlog. So the most likely explanation is that something is keeping the engines too busy to get the connection handler scheduled before your connection attempt times out, or to work on the internal timer from the deferred queue.
Is this server involved in any replication using dbcc logtransfer replication (as opposed to Rep Agent replication)? There is another bug fixed in 15.7 SP120 that results in high CPU use under that condition, which might result in your symptoms, CR 745325. If you aren't doing any replication, don't worry about it.
We would normally try to look at cpu busy issues with sp_sysmon, but if you can't log in, you can't run sp_sysmon. The next thing I'd try using to diagnose this would be a couple of manual sybmon memdumps taken perhaps a minute apart while this was happening. For directions on how to take a manual memdump, please see https://support.wdf.sap.corp/sap/support/notes/1940109. You should open a support instance to have the memdumps looked at.
-bret
