Postgres will not allow new connections, suspended process, waiting error

We have a recurring problem where Postgres will not allow any new
connections. Any connection already made keeps working fine. Every time
this happens, we also have one suspended postgres.exe process that can not
be cancelled. Also, we get the following error in the logs every minute:
‘WARNING: worker took too long to start; canceled’.

We’re using Postgres 9.5 on Windows Server 2012 R2. Our original hypothesis
was that there was something wrong with the autovac settings, but we copied
the settings over from our other server that does not have this problem and
still continue to have problem.