What you describe (broken wire in runs) is possible. The horse fence I have up started out as a wad of electric tape the previous owner left when we purchased the place. Strung up where I wanted it, I had a good hot fence when it left my connection to the fencer, but none at the terminal end. So somewhere along a few hundred yards of fence it stopped working. (BTW, if your terminal end is by the starting place, it makes for quick testing. If it's hot at the beginning and hot at the end, you can be assured it is hot all the way in between.
So to trouble shoot something like that you use your fence tester. That fence had about 5 or 6 splices in it, but as I went along, the connections were good and fence was hot. Got about 3/4 the way around and at the final splice it stopped working. So I checked on the other side of the last splice (the hot side) and it was not working there either. So backing up, I eventually concluded the splices were all good, but somewhere in the run it had gone dead. So testing every 10 feet or so, I eventually isolated the problem and found the dead spot. It was broken wires in the middle of the run and it looked no different than any of the rest of it. Considered my options, I eventually decided to purchase new tape to splice in from the dead spot to the terminal end. That worked and the fence lit up and was hot coast to coast.
BTW, for those who would like to test your fence, but don't have a tester, the crude but effective old school way of doing it was to take a screwdriver or some such tool with an insulated handle, ground the shaft of the screwdriver to a metal post or something similar (something you know is grounded) and while touching the ground, then touch the tip of the fence to your hot wire. You should see and hear a spark. A hot fence will spark pretty good. A weak one, very little to none at all.
Or.........grab it. To anyone who might wonder how effective a really hot electric fence can be, you really ought to experience it just once*. Touch a hot fence with your hand and odds are will make a believer out of you.
* If you have a heart problem or pacemaker, you can take my word for it.