It is in fact NTFS permissions, and a strange bug in SQL Server. I'm not sure the above bug report is accurate, or may refer to an additional bug.
To resolve this on Windows 7, I ran SQL Server Management Studio normally (not as Administrator). I then attempted to Attach the MDF file. In the process, I used the UI rather than pasting in the path. I noticed that the path was cut off from me. This is because the MS SQL Server (SQLServerMSSQLUser$machinename$SQLEXPRESS) user that the software adds for you does not have permissions to access the folder (in this case a folder deep in my own user folders).
Pasting the path and proceeding results in the above error. So - I gave the MS SQL Server user permissions to read starting from the first directory it was denied from (my user folder). I then immediately cancelled the propagation operation because it can take an eternity, and again applied read permissions to the next subfolder necessary, and let that propagate fully.
Finally, I gave the MS SQL Server user Modify permissions to the .mdf and .ldf files for the db.
I can now Attach to the database files.