I installed ArcGIS 10.2 on a fresh (virtual) machine. I copied some existing File Geodatabases (version 10.1) via network share on the new machine to a local drive. Some File geodatabases can not be opened:
Failed to connect to database. Workspace or data source is read only. Workspace or data source is read only. Workspace or data source is read only.
All following tries will add more "Workspace or data source is read only." messages to the error dialog.
Do you still have a machine with v10.1 on it? Try to use the same file geodatabases on that machine to see if they still work. Maybe there could be some corruption.
I think what you might be experiencing is NIM090672: Specific Microsoft updates may result in data corruption with file geodatabases or shapefiles created on a network share. In short, when writing data to a file geodatabase on a network share using a few select Windows operating systems, errors can occur upon subsequent request of that same data. In your case, this looks like it could be the issue based on what you reported. Check out the link below to determine if this is correct in your case and how to apply the fixes:
