They are both related.
From what I can recall bluehost limits connections to mysql to 30 simulaneous connects. All it takes to trip that limit is 31 people, spiders, bots, whatever to access the site at the same moment in time. That accounts for the first error.
Also, bluehost had (don't know if they still do have) a 60 second real-time CPU usage limit over a 3 minute period. That can account for the second error.
Those are simplified explanations, but close enough for this purpose.
You can lower the connections a bit by not allowing spiders and unregistered users access to things like the calendar, who's online, member profiles and any other info that might be useless to them.