  We explore the sensitivity of $^{163}$Ho electron capture experiments to
neutrino masses in the standard framework of three-neutrino mixing and in the
framework of 3+1 neutrino mixing with a sterile neutrino which mixes with the
three standard active neutrinos, as indicated by the anomalies found in
short-baseline neutrino oscillations experiments. We calculate the sensitivity
to neutrino masses and mixing for different values of the energy resolution of
the detectors, of the unresolved pileup fraction and of the total statistics of
events, considering the expected values of these parameters in the two planned
stages of the ECHo project (ECHo-1k and ECHo-1M). We show that an extension of
the ECHo-1M experiment with the possibility to collect $10^{16}$ events will be
competitive with the KATRIN experiment. This statistics will allow to explore
part of the 3+1 mixing parameter space indicated by the global analysis of
short-baseline neutrino oscillation experiments. In order to cover all the
allowed region, a statistics of about $10^{17}$ events will be needed.
