Yeah there's special tech for that! Has something to do with sending some frequency over the line and measuring how long it takes to "reflect" off of the breakage and back to the shore.
In the non-optical domain, that used to be a feature in the most expensive cable measuring gears (to the tune of €25k about 20 years ago).
I had the pleasure of using one of the devices that had that when I wired my old house. Borrowed the kit from ... an institution that allowed me to get their gear for a weekend. (Yes, I knew people in there.) Having the device tell me that a given strand in a CAT5e cable was faulty at <this many meters> from my location made it surprisingly easy to detect where I'd messed up.
I was going to say that this was invented by none other than Heaviside, but it turns out his technique relied on static resistance measurement to find the cable fault. Similar concept though. And in 1861!
This video explains and show the same concept for copper cables in a very simple and hands on way:
"Cheap and simple TDR using an oscilloscope and 74AC14 Schmitt Trigger Inverter"
https://www.youtube.com/watch?v=9cP6w2odGUc
Some NICs and managed switches can do it as well. Well, I should say that most can do it (as in the hardware is capable) but only some actually expose the functionality.
These used to be big, expensive, standalone devices (I'd occasionally use one in a previous job). Nowadays, there are portable, handheld units specifically for identifying where a break has occurred and the same, basic functionality is often even built into the gear used at either end of connections (e.g., even the "enterprise" switches -- from Brocade and Cisco -- that I use at home).