The next morning I took a look, and some sort of critter has been chewing through the wire! They completely chewed off three bulbs right at the socket, leaving the socket on the ground, and nearly got a fourth chewing a few inches away from the light, which makes me think that they chewed through the strand and then chewed off the wire on the ground.
I'm not exactly thrilled with the idea of twisted wire and electrical tape splices outdoors, but we'd only use the lights when it's patio weather out and we're present, and it's always just one conductor, so I think it'll be okay.
But I'm stumped as to why critters would be chewing off light bulbs and eating the wire!