Good for you for thinking ahead! We also have special funds set aside strictly for animal emergencies. With three senior horses, a doberman, and a cat...sooner or later I'm sure we'll need it. I would rather have it and not need it, then need it and be stressing over how to pay for it!
Great idea the only thin I thought with saving accounts is that if something happened next week, there might not be enough to cover it. Also, life long conditions can eat up savings money very, very quickly. I decided to do both; have a small savings account and the best pet insurance I could get. My pet insurance is £50 a month, which is about £600 a year, which I figure will be about £6500 over his lifetime (aprox 10,400USD). The uetion I asked myself was 'is he going to cost me more than £6500 in vet bills in his lifetime?' probably, with all the breed problems we unfortunately have. luckily my insurance company are very good too, and won't stop costing him when he he's old.
We don't have pet insurance but instead put money away every month into the emergency fund. Thankfully over the past few years we haven't had to use it so it's grown to be quite a lot...hopefully more than enough to cover most any emergency that should come up. We pay for routine vet visits out of our regular account so that emergency fund is never drained.
I keep a separate checking account for my dogs. Show expenses, breeding expenses, health testing expenses....etc... come out of it. It can also be used for dog emergencies. I pay normal dog care out of the regular family account along with normal vet bills not related to breeding or health testing.
With three personal dogs, the cost of health insurance is pretty cost prohibitive.
I'm pretty fortunate that we have enough money to cover emergencies if needed.