The Userids in question on both computers are confirmed as full Administrators.
Anyway, the plot narrows down a tad.
I now know that it is only files saved in System32 via a web browser that behave as described. Files saved via Windows Explorer remain fully visible at all times.
Both computers are using Symantec Endpoint antivirus v11 so I'll have a poke around in here looking for web protection settings that might be doing it. If they are, I'll probably leave them set once I have proven the cause - It's just that having seen the interesting behaviour, I want to know what causes it.