Are you sure about that? You may be thinking only of conventional attack subs, but don’t forget that subs have gone to the bottom of the Mariana Trench (depth - over 36,000 feet). It doesn’t take a huge conventional sub to cut a cable, nor does it even have to be manned.
You are correct - I was thinking of a conventional military sub. Using a deep sea submersible 8,000 ft would not be beyond range, especially for an unmanned unit.