We started getting travel insurance a couple of years ago, after we got on Medicare and realized that overseas coverage is limited or non-existent. I found policies from all the major players (Allianz, AIG, and even through Squaremouth) to be very expensive for the coverage provided. Eventually found World Nomads, and have been using them ever since, even for things like cancellation insurance for trips here in the US. They typically offer two tiers of coverage, and for us even the lower tier is sufficient.
We have never had to file a claim with them, but when I contacted them for an issue we had in Coimbra, Portugal, they were responsive. The reason we didn't have to file a claim was that the hospital told us not to worry about it and have a nice day. And that was after seeing a very attentive doctor, X-rays, a nurse, etc. You know, the kind of thing that would have probably cost over $2K here in the US.