Broker becomes unresponsive

I note that frequently the response given to broker inquiries is that the broker is not likely to be a source of performance problems. While conceptually, I tend to agree with this recent circumstances make me wonder. I would like to entertain a discussion along the lines of ‘What WOULD cause a broker to become unresponsive?’

Recently, I have observed the following errors:
Timeout (112-1450): The request timed out
Timeout (112-1452): Initializing a connection to host

These occur when a client app is trying to do something relatively mundane like check a canPublish or canSubscribe permission. Other connections and operations on the broker are succeeding (ie this is happening sporatically). There does not appear to be any significantly higher volume at the time that the errors are reported.

Any thoughts?