We show that the periodic boundary conditions typically applied in numerical simulations of vortex dynamics can lead to significantly incorrect results even when the vortex cores are small compared to the computational domain. This is demonstrated for two previously studied flows which capture significant flow physics: (i) an isolated vortex embedded in fine-scale turbulence; (ii) two antiparallel vortices of unequal strength undergoing reconnection. In case (i), periodicity, when invoked, results in strong, unphysical turbulence growth leading to vortex core transition, whereas the vortex remains totally intact during its interaction with the turbulence when periodicity is not invoked. In case (ii), the vortex interaction, including reconnection, is significantly distorted. These differences are due to the artificial zero circulation constraint, inherent in periodic simulations.