Sometimes when a WordPress blog gets put up on digg.com or slashdot - the MySQL server that is storing the data gets overloaded and then WordPress can't establish a database connection. Many servers don't have this problem - but the few that do bring up the wordpress "Error establishing a database connection" message.
The problem with the screen as it exists now is that it has a WordPress logo very big on it - is called "WordPress > Error" - both things make it seem like it's WordPress's fault that there is an error. This is obviously not the case - since it's MySQL's fault.
I think the page title should be "MySQL Error" and the WordPress logo should be made much smaller on the error page... Perhaps a MySQL logo could be put on that page to make the visitor realize it's not WordPress - but MySQL that's at fault.
The reason I say this is because in the comments on articles once this happens many people will say stuff such as "Man, WordPress really sucks - it can't handle traffic..." - so you can see that the error message is doing a lot to damage the reputation of WordPress...