Well you have to remember that guests and bots such as google are going to generate activity on the server. I am no expert but even with thousands of web server connections and hundreds of users I only observe around 100 actual mysql connections at the most. But the limit on a shared host may be less than that and in fact that many connections is probably not needed for a small site. But it is something to look into.
Basically this error was returned by the mysql server and is directly related to the max connections. However if the number of connections is abnormal for the site the cause may need to be found.