Connection limits to MySQL are under the hosts control. Unless you have access your self to editing MySQL's configuration, specifically the entry below:
# The maximum amount of concurrent sessions the MySQL server will
# allow. One of these connections will be reserved for a user with
# SUPER privileges to allow the administrator to login even if the
# connection limit has been reached.
max_connections=255
HOWEVER it may not be all up to your host 
You may be able to cut down on the number of connections your board generates by looking at the plugins you have loaded and also the options you have turned on on your forum.
I'd start by turning on via the setting in /include/common.php
// Enable DEBUG mode by removing // from the following line
define('PUN_DEBUG', 1);
to see how many queries each of your page views is generating.
Without registering to see your forums, it is possible that you may have installed plugins that are spawning excessive MySQL queries. This is certainly not unheard of - eg in the case of the various sub-forum mods, which are very inefficient and spawn *large* numbers of MySQL connections per page view.
Even some of the standard punBB features like the 'Users online' option that shows all the users currently online generate an extra MySQL load per page view. Perhaps you could review your use of some of those extra punBB options.
By optimizing how many queries your pages spawn per user page view, you may be able to cut down the number of simultaneous connections MySQL wants to open, and operate within the current limits set by your host to your usage of MySQL.
