We observe apparent hole pockets in the Fermi surfaces of single-layer Bi-based cuprate superconductors from angle-resolved photoemission. From detailed low-energy electron diffraction measurements and an analysis of the angle-resolved photoemission polarization dependence, we show that these pockets are not intrinsic but arise from multiple overlapping superstructure replicas of the main and shadow bands. We further demonstrate that the hole pockets reported recently from angle-resolved photoemission have a similar structural origin and are inconsistent with an intrinsic hole pocket associated with the electronic structure of a doped CuO2 plane. © 2011 American Physical Society.