“using stale statistics instead of current ones because stats collector is not responding”
To temporarily alleviate the problem until you find a solution, have you tried sending a SIGHUP to the server when it occurs?

I have a similar problem in a fairly different situation: the database runs on a windows desktop PC together with Apache and proprietary machine control software. My case fits in the "slow or overloaded" category. However, I see this happening even at night while the machine is idle. I've been monitoring many system parameters with PRTG to try to find a link with a specific activity, but I can only see the consequences, not the cause.

I don't see this as a simple warning because, once it starts, queries that normally take 100ms will start taking 10 seconds (but still leave other queries unaffected). Also, it stays that way until I send a SIGHUP (pg_ctl.exe reload ...), which I think is weird since the process should resume once it is no longer starved.

P.S. please don't hail me about the windows desktop part: I know, and ... circumstances.