For reasons that are not entirely clear yet, I started to get a Event ID 424 on a SQL Server Analysis Services machine in the Windows Log/Event Viewer (Application) and Event ID 23. ID 424 is about an invalid encryption key and 23 that a cryptographic key could not be loaded. The only entry on MSD Social had suggested (until I added my reply) that a reinstall was required.
Fortunatly it is not, you just need to rename the current OLAP data directory and create a new blank one of the correct name, whilst SQL Server ANalysis Services is shut down of course. Once SQL Server Analysis Services is restarted all should be fine. Though you will need to recreate all your old cubes.
At the moment the potential root cause for the problem I saw was enabling OS level encryption on the data folder for SQL Server Analysis Services, so it may not be a good idea to try that. Once I have tested that theory I shall update this post.