I'm posting this here as it's the best way I can think of to give the info a chance of being seen by someone at 2degrees who might be able to do something about it, and of helping anyone else who might hit the same problem I did.

On the registration page for creating a new Your 2degrees online account - the page to which a user is sent after clicking the link in their verification email - the user is prompted for four pieces of information:

First Name
Surname
4-digit PIN
Alternative Phone Number

The form checking on the web page itself will reject 4-digit PINs that it considers too obvious e.g.

1234
2222
9876

with a message saying "This value seems to be invalid" (which itself isn't the right error message, but anyway).

However, it appears there are other PINs that the front-end will allow to go through, but which the back end will reject. But there is no error handling for this, so the user gets the catch-all error which says:

"Things aren't working as they should so we can't complete your request. Please try again or call us on 0800 022 022 if you continue to have problems."

An example of such a PIN is 0110.

And yes, I found this out the hard way. Took me a week. Yes, I know it's a really bad PIN. 

From the various phone calls I've had with 2degrees telephone support I can confirm that it's not something they're aware of, and I don't much fancy the chances of my successfully explaining it to them, even less of that explanation then successfully wending its way through the system, hence this post.