Update finally. I bled pressure, then put about 42 psig in each tank. While I had the pressure down, I disconnected the pressure switch and checked both the connections where it joins to the pump discharge and the switch itself. No blockage or debris. Some corrosion, but no way it is causing a bad reading. I did not replace the gage because it is fairly new and tracks consistently with what I expect. I adjusted the cut-out up to about 65 psig, cut-in is now at about 47 psig. I still have no problems at my house.
My neighbors have had their water go out now three times in the last week. My neighbor observed the gage for one time and I did for another. Both times, we saw around 50 psig on the gage when they had no water and I did have water. Then we would turn water on at my house, the pump would come on around 47 psig and when the pressure rose to about 53 psig, they would get water back. As I see it, at the low end of the operating pressure range, there is an occasional event that causes no flow of water to my neighbors house. It doesn't happen all the time, but when it does, they don't get flow back until 53 psig or so.
It is not a slow return of pressure to their house; it goes from no flow at all to normal flow when it comes back on. Note from previous posts that with continuous water on at their house, and turning power to the pump off, they have flow all the way down to 25 psig at least, slowing, but still flowing.
So, my plan now is to check the pressure on the tanks in about a month (after turning off the pump and going to no flow at my house). If there's a problem, I'll replace the tanks.
I still think their problem with water flow is unrelated to that. Still seems to me that a flow restriction in their line exists, and since it seems to be a "sudden on" as pressure comes up, a sticking check valve seems logical. Should I tell him to start digging where the line comes in to his house or at the outlet to the pump house? Other thoughts?