Do you get this error report often, once in a while or is this the first ever? I get ocassional error reports like this or similar. As long as it doesn't happen all the time it's usually not a big deal. Many times it can be attributed to your host. Maybe they are having some issues or problems. Maybe something is up with the mysql server that they are working. This could easily cause this.

You could ask them what the max mysql connections are per account. It's a fairly common question for a host. see more Like I mentioned most usually have some kind of limit per account, 20-30 maybe even 50.

If you visit your site often times you will make more than 1 connection depending on what you are doing. Maybe 3-5 connections briefly then it will drop down to 1 or 2 and if you are idle eventually maybe even 0, until you refresh or click on something else.

All I could recommend if this is the first time, don't worry too much. If you start seeing in more often and/or your site is seeing more traffic you might check with your host as mentioned. If they have it set low, you could try another, but you don't want to move your site to another if you don't have to. It is pretty easy but a little confusing the first time.

Finally there is a phpinfo file available in version 6.1x to check some of this. Just go yoursite.com/admin/phpinfo.php you will be prompted for you admin login.

Scroll down to the mysql area about 1/2 down the page. You will find persistent and active links/connections at the moment, also something like the following:
mysql.allow_persistent On On
mysql.connect_timeout 60 60
mysql.default_host no value no value
mysql.default_password no value no value
mysql.default_port no value no value
mysql.default_socket no value no value
mysql.default_user no value no value
mysql.max_links Unlimited Unlimited
mysql.max_persistent Unlimited Unlimited
mysql.trace_mode Off Off

Possibly mysqli under it. Depending on these numbers and values you can increase some with .htaccess or php.ini depending on hosting type.

Just a few thoughts.