I've done further testing. A backup is currently running from the ReadyNAS to my PC, with about 60,000 files processed out of about 236,000. The NAS was accessible from the PC which is running the backup software until it got past about 16,000 files, then it became inaccessible. I used Windows Resource Monitor to look at the "Associated Handles" of the process which is running the backup. It appears that there are about 16,384 (2^14) handles for open files. I can see the directory names in the handles, which are nearly all in the first 16k files the backup would encounter.
So it seems that Samba on the ReadyNAS is not closing the files to the satisfaction of Windows 7 x64. The backup continues after that though, so these files have all been processed. If I stop the backup mid-way, then the NAS remains inaccessible until I close the backup program on the PC, killing the process, which releases the open files. Then I can access the NAS again.
Again, what version of Samba is running on x86-based ReadyNAS machines? I suspect that it is using an old version of Samba before the fix was introduced in Samba 3.5.0. Here's the change log again (search it for BUG 6837: "Too many open files" when trying to access large number of files from Windows 7):http://www.samba.org/samba/history/samba-3.5.0.html