It's definitely a web server/database issue and there's nothing we can do about it as humble web browsing mortals. The forum script (SMF) is saying it can't make a connection to the database, nothing to do with the end user's web browser.
The SMF help page at
http://wiki.simplemachines.org/smf/Repair_settings.php says the forum software can be set to use standard or persistent connections, I would guess that it's set to persistent at the moment and there are too many lingering unclosed database connections or mySQL doesn't have enough available connections.
