You do need to look carefully at what they cover and don't cover. Most won't cover a pre existing condition, genetic conditions (more so with dogs, such as hip dysplasia), or routine procedures such as vaccinations, spay/neuter or parasite treatments/prevention. Depending on the policy, other things might not be covered or there are limits on how much they will pay per pet or pet condition. Each company is different and many have a few levels of coverage. Many companies don't cover small animals, so you need to check that as well.
Having your own account for vet bills is a good idea. Even putting $50 a month can really help when you need it.
There is also Care credit. It is like a credit card, but can only be used at vets, doctors and other medical places. I think there is no interest for 6 months and after that it is still pretty low. I don't know if it is just a North America thing, but is worth checking out.