In previous work we have shown that aligned high aspect-ratio (pseudo-2D) nanofillers can yield large dielectric breakdown strength (EBD) improvements for a nanocomposite with a low-crystallinity polyethylene matrix. Here, we report a systematic study which delineates the contributions of the aligned inorganic fillers and of the aligned polymer crystallites in the overall EBD improvement achieved in the nanocomposites. Specifically, extrusion blown-molded polyethylene/montmorillonite nanocomposite films were cold-stretched to various strains, to further align the nanoparticles parallel to the film surface; this filler alignment is accompanied by a commensurate alignment of the polymer crystallites, especially those heterogeneously nucleated by the fillers. A systematic series of films are studied, with increased extent of alignment of the fillers and of the crystalline lamellae (quantified through Hermans orientation order parameters from 2D X-ray diffraction studies) and the aligned structure is correlated to the electric field breakdown strength (quantified through Weibull failure studies). It is shown that aligned pseudo-2D inorganic nanofillers provide additional strong improvements in EBD, improvements that are beyond, and added in excess of, any EBD increases due to polymer-crystal orientation.