I was out walking my dog without a leash (which I would never do now) and my neighbor's dogs got out and chased my dog, jumping on her and breaking her leg. I asked the owners of the dogs to pay the vet bill and they refused so I took them to court. We were both at fault because neither one of us had our dogs on a leash, I was awarded half of my vet bill and court costs. I thought that was fair.
Cats are free roamers and stuff happens to them. I personally don't keep my cat indoors, my last cat was 18 when she passed away, so I have pretty good luck with my cats being indoor/outdoor. They have a cat door in the family room and in the garage so they can get in quickly if they need too.
If my cat were hurt by a dog off the leash in neutral territory I would pursue help with the vet bill. If my cat were hurt because he was in my neighbor's yard and the dog got him, I would assume all the responsibility of the vet bill.