Issue
When trying to SSH into my dedicated database servers, I obtain a Permission denied error message
Cause
This is actually the normal behaviour on Acquia Cloud. We don't provide direct SSH access to the database servers, and you can only connect to your web servers via SSH. It is also not possible to SSH into the balancers and into the svn servers.
Resolution
You can administrate your database via a third-party application, as explained in the documentations SSH tunneling for server-side application and Connect MySQL Workbench to an Acquia Cloud database