Welcome to the forum. Before we can try to assist you, we are going to need a LOT more information from you. What Mac and what version of the OS are you running? (Click on the upper left corner of screen, then "About This Mac" and post what you read here.
As for how Time Machine works, the first time it makes a backup it backup up EVERYTHING you tell it to. After that it makes what are known as incremental backups, that is, it only backs up what changed since the last backup. To keep it all in order, when it finds a file or folder with zero changes, it creates a small pointer to that unchanged file in the previous backups. So the second backup is much smaller than the first. And after a while it deletes most of the backups and keeps just one for that day, then it deletes more later to keep one for that week/month. Here is Apple's article on Time Machine:
https://support.apple.com/en-us/HT201250
That approach means that the TM backup only grows very slowly. So, to answer your question, is it possible to max out a 1 or 2 tb drive? Sure, but it will take a very long time to do that. And you don't clear it out, TM does. If you go mucking about in a TM backup through Finder or some other non-TM application, you will destroy the integrity of the backup and the chains of pointers to files that haven't changed in a while.
But give us more about how your backup "stopped working" and we can see what is happening. How do you know it's stopped? What settings are in Time Machine in the System Preferences? How full is the drive? How is it attached to the Mac? How often is it attached to the Mac?
