We analyze the energy and zenith angle distributions of the latest two-year IceCube data set of upward-going atmospheric neutrinos to constrain sterile neutrinos at the eV scale in the 3+1 scenario. We find that the parameters favored by a combination of LSND and MiniBooNE data are excluded at more than the 99% C.L. We explore the impact of nonstandard matter interactions on this exclusion and find that the exclusion holds for nonstandard interactions (NSIs) that are within the stringent model-dependent bounds set by collider and neutrino scattering experiments. However, for large NSI parameters subject only to model-independent bounds from neutrino oscillation experiments, the LSND and MiniBooNE data are consistent with IceCube.