Partial answer: when JMP opens a table, it tries to get a write lock to prevent another user from also updating the file on disk (because one of the two users would lose their changes.) The lock is released when the table is closed. I think you'll see the "locked" in italics in the left panel when JMP can't get the lock.
It sounds like this is happening within a single instance of JMP. I'm not sure why JMP wouldn't still see the network disk file as the same and use the same already open data table.
You might do an experiment with "unc" names, something like
\\server\path\xxx.jmp rather than the drive mapping to L: and see how that changes things.
And what was the previous version of JMP that worked?
edit: make the unc name a little more correct
Craige