I have been growing potatoes for over forty years.
I must have been real, real, real sloppy or lazy last year when I dug them up; now I have mentioned the Purple Peruvian invasion earlier but in both garden this year, even where the PP were not planted I have volunteer potatoes popping up all over the place.
One year I did miss most of a whole row but that was all, I have had one or two odd balls on occasion but this year if I had not planted any I would have still plenty potatoes.