Firefox 29 has a totally new way to Sync that is much easier and reliable than the old method. You do need to have all your machines on Firefox 29, so go ahead and update them all now.

Then read Sync your Firefox bookmarks, history, passwords and more

Thank you for your answer, but this doesn't solve my problem. The thing is that I cannot access my old Firefox (it was version 27 or 28), because I had to reinstall Windows on that laptop where the that particular Firefox was running. So I cannot upgrade it to Firefox 29 or unlink that device. Because I have reinstalled the OS and the old device doesn't exist in that sense. And when I tried to log in to my Sync account on the reinstalled Windows (on the same physical laptop) using my Sync account credentials, I got the error I mentioned before.

Why do I get that error message if my user account credentials are the same and are correct? How can I retrieve all my bookmarks and passwords from my Sync account that I created with my older Firefox?

Tyler posted a link to the instructions for updating to the newSync. If you can't follow those instructions due to an OS re installation, I don't know how we can help you.

Do you have any other machines connected to that Sync account?

If not, Sync wasn't intended to be used as a "cloud" for one machine. The system and the upgrade process to the new version of Sync isn't set up to work as you are trying to use it.

I have the same problem and I really need those bookmarks.

How can we get the bookmarks from he old account?