Well.
I took the modem and computer to a T-mobile outlet. My modem produces the same fault whichever computer it is plugged into. Other, apparently identical modems don't show any problem. This seems to identify the problem (though God knows how) as being the modem.

But.
The contract was signed in a branch of PC World who also supplied the modem so the T-mobile shop told me to go to PC world. At PC World we went through the whole thing again resulting in many "but this is impossible" remarks (which I actually agree with) and the answer that T-Mobile no longer supply this model so I need to get back in touch with T-Mobile.

So.
I returned home and called them up and went through it all again with a different (though still helpful) technician resulting in more "but this is impossible" remarks.

Result.
They will send me a secure packaging envelope to return the modem to them, leaving me without an internet connection at all for a while.

But.
They won't try to fix it, they will just do a straight swap of both modem and sim card which MAY (only MAY) fix the problem.

Therefore.
If I disappear altogether for a while don't worry. I shall return.

(As ever I may not be able to see replies to this post.)

Well, progress of a sort, at least. This could become very interesting (if schtupid). Perhaps you should start a blog detailing your daily struggles?

Now here's weird!

How in a million hells can it be the sim card!

I haven't had the packaging to send the modem back yet but I have received a replacement Sim card. I've put it in and as far as I can tell from five minutes checking the problems have gone away. This should be plain impossible but there you go.

For the moment, touch wood, I seem to have access again.

Of course as I'm off to a beer festival this afternoon I probably shan't be posting anything now today (and who knows if I'll still have access tomorrow.

Seriously though, how can anything related to the sim card possibly cause the symptoms that I've seen?

Anybody out there with enough tech knowledge to even form a theory on this?

I have absolutely no idea, but welcome back, Bob! I just saw your posting on your blog about it. Maybe you'll be able to take part in the chat now? Even if you're off to the festival this afternoon.

The only thing I can figure is the insertion of the SIM card reset the digital modem. I was about to ask you if you can connect to the modem with a web browser. It might be some funky setting in its configuration, like some kind of filtering. Anyway, I hope your problems are over.

Seems I was premature.
The problems are back again, exactly as before. I'm back in touch with the support team.

I'll update you when I have news.