Now this is REALLY weird.
I copied the .mht files onto a pen drive (which is how I tried them on the old XP/IE7 laptop). I then thought I'd better also try them out on another Win7/IE11 machine. So I plugged the pen drive into my Win7/IE11 laptop - no probs.
So back to my Win7/IE11 desktop (where I'd had the problem). Plugged the pen drive into that machine and opened .mht files in IE11 without issue. BUT, BUT, BUT trying to open the very same .mht files direct from the hard disk in that machine (and from where I'd copied the files to the pen drive) gives the IE11 error. So what the expletive deleted is happening????