Can I run Drush in a situation where the database is on one server and the remainder of the Drupal installation is on another? I have a setup (out of necessity not choice) where my drupal installation is on server a, and the database on which it relies is on server b.  The site is working fine, however when I run most Drush commands I get the good old:

Drush was not able to start (bootstrap) the Drupal database.

On the server where everything except the database is stored, there is no mysql.  If drush isn't an option for me that's perfectly fine but I've been tearing my hair out on this one.

Many thanks in advance for any ideas.
I've never run a Drupal site like this, but Drush should take your database settings from your settings.php file; if the database is remote for Drupal, then it should be remote for Drush as well.  Note that most Drush commands (with the notable exception of the sql-* commands, which you will not be able to use without a local mysql installation) just call through to the corresponding Drupal code, after first bootstrapping.  If your bootstrapping is failing, check to see if there is anything in your settings.php that depends on the site's URI to select the database.  If so, make sure that you use the --uri= flag, or set the 'uri' element in your Drush site alias when using Drush.  If that doesn't get you going, post the output of a failing Drush command with the --debug flag specified.