Wordpress - Is there a plugin that will override the "Error establishing a database connection" message?

http://yoast.com/custom-wordpress-database-error-pages/

You can make your own Database Error page by adding a db-error.php to your wp-content folder (/wp-content/db-error.php). You can find a good example of such a page in the link above. Don't forget adding header("HTTP/1.0 500 Internal Server Error"); in that file so it get a proper header message.

I'm not aware of such a plugin, and will be happy to upvote any appropriate answer.

On the other hand, a way not to display this message is not to have the error any more. On most hosting providers, connections are refused when the maximum number of open connections is reached.

A cache plugin (such a WP Super Cache) will drastically decrease the number of connections and make this error message less frequent.