You mean hosting your application on Windows Azure and then hosting the database on your own servers? This is totally possible, but you'd have to publicly expose your database. You could host your database on SQL Azure and that would be a 'separate server' from your Windows Azure host. With SQL Azure, you can set up a firewall rule to allow only your application to access your database.
If you were to host your database on your own servers, I don't think that it would completely diminish the value of hosting your application on the cloud as your app tier at least would still be hosted "in the cloud". Though, you could also run into performance issues in this scenario as SQL is a little slow to access over the internet using TCP.