Initially I suspected this was related to the Wordpress framework I'm using (Genesis) or to my child theme functionality, but turns out that's not the case as I was able to get this to work in a test environment running on the same framework and child theme.
Possibly this is related to the caching plugin I'm using, W3 Total Cache. If this plugin is not active, then the custom db-error.php file is picked up.
To rephrase my question, has anyone run into this issue with caching plugins being active?
Has anyone successfully been able to replace the default Wordpress database error page with a custom one?
I tried this (add db-error.php under wp-content), but I can't get it to work for some reason: http://css-tricks.com/snippets/wordpress/custom-database-error-page/