Figured this one out. The terminals on the fixture are not the end of the line, as there is a slide contact that can corrode. There's a little tab on the fixture that you depress upwards, then you can slide the whole thing out and see the contacts. One of mine was corroded. I wet sanded it with some electrical contact cleaner, and all is well. See this eBay picture; the one for sale sadly has both slide contacts badly corroded:
