The dynamic aperture of the HERA proton ring is limited by persistent current field errors in the superconducting magnets. Its value as predicted by tracking studies is compared to preliminary aperture measurements in the machine. Since inclusion of tune modulation leads to a significant reduction of the aperture in the simulation, a theory is developed, which quantifies the impact of tune modulation on high‐order resonances. For a typical working point, parameters of resonances up to order 11 are calculated for a model of HERA using differential algebra and normal‐form algorithms. Based on these resonance parameters, estimate values are derived of amplitude‐dependent emittance growth rates caused by the interplay of tune modulation and nonlinear field errors.