The central tenet of effective theory is that the details of short-distance physics will not have a significant impact on low-energy observables. Here we perform an analysis of electron-deuteron scattering at low momentum transfers which is based on effective field theory. We show that in our approach the deuteron electromagnetic form factors F C and F M indeed are largely insensitive to the short-range NN potential at momenta Q up to about 700 MeV. We also find that the effective field theory approach to deuteron electromagnetic structure provides a systematic justification for many features which have been seen in potential model calculations of the same quantities.