According to this official Tweet, when I write a new tweet, I can choose an old tweet to attach to. But I couldn't figure out where this feature is on desktop browser.
This document says the first step is:
Tap the compose icon.
But I couldn't find such an icon either.
You will notice that this is a business account, and that it is "verified".
One of the features of verified accounts is "Allow contributors so multiple people can tweet from your account".
You can find out more on twitters own website, but this service is in beta and is not generaly available yet.
I'm using Twitter for business. Can I verify my account? Not yet. Twitter is alpha testing verification for businesses and we hope to have more information soon. Check our business section for more updates!
This servie is due to be rolled out this year I think...
I was getting the same error message too.
Clear your cookies and log in again - it should work then. I have provided a screen shot just so you know I am not making it up :)
EDIT
After a bit more digging it appears that there is an issue with Twitter. You can switch back to the old twitter but you have to use the link that appears in the blue banner at the top of the page.
If you try to use the option in the drop down called Switch to Old Twitter it won't work.
I think the reason for this is that it executes a HTTP POST with some hidden form values. One of these values is the authenticity_token. This value is not set for the drop down option. However, looking at the link in the banner, the value is set.
I suspect this is what is causing the problem.
By clearing your cookies your are forcing Twitter to display the banner which has the correct hidden form values set. This is what enables you to view the old twitter.
EDIT 2
This issue has now been fixed by Twitter and you can simply switch to the old version of Twitter using the drop down menu option.
Best Answer
The feature you're referring to in the first link you provided in your question is available only on Twitter mobile. To add or thread a tweet to an old tweet all you need to do is to reply it. Technically speaking, the feature mentioned by Twitter is simply bringing an old tweet to view based on context while re-packaging the reply button. On a desktop you can do it yourself by finding the old tweet you wish to thread a new one below it, and then simply click the reply button on it to attach a new tweet to it.
Even if your old tweet had already accumulated lots of replies, if it is you who reply to your own old tweet, Twitter will thread your new reply tweet at the bottom of the old tweet and not in its chronological reply order.