Materials with increased functionality are often based on crystalline structures with significant local disorder. Typical examples are ferroic oxides exhibiting large spontaneous polarization that can be rotated by an applied electric field, finding use in many important applications. Despite years of investigation, the exact structural origin of the increased piezoelectric response of oxide ferroics is still unclear. Frequently evoked models attribute it to emerging polar nanoregions inside a nonpolar matrix, the existence of a morphotropic boundary separating polar phases with different crystallographic symmetry, low-symmetry bridging phases facilitating polarization rotation, and displacive and order-disorder structural phase transitions. Here we use both conventional and resonant high-energy x-ray diffraction coupled to atomic pair distribution function analysis and three-dimensional computer simulations to examine the relationship between the local structure and piezoelectric properties of exemplary sodium-potassium niobate ferroics. We show that their increased piezoelectric response is primarily due to a geometrical frustration in the underlying perovskite lattice induced by local fluctuations in the tilt pattern of the constituent niobium-oxygen octahedra, and not to a crystal-crystal phase transition or distinct nanodomains. The fluctuations peak when the sodium to potassium ratio approaches 1, leading to a softening of the perovskite lattice and easing of polarization rotation under an electric field. Based on the experimental data and model calculations involving Goldschmidt's tolerance factor for the stability of perovskites, we also show that the fluctuations are driven by the mismatch between the radii of sodium and potassium atoms, and the increased piezoelectric response of sodium-potassium niobates indeed scales with the variance in the distribution of these radii about the average value. Thus, we settle important aspects of the debate over the structure-piezoelectric property relationship for oxide ferroics, thereby providing a different perspective on the ongoing effort to improve their useful properties.