Axion dark matter (DM) may convert to radio-frequency electromagnetic radiation in the strong magnetic fields around neutron stars. The radio signature of such a process would be an ultranarrow spectral peak at a frequency determined by the mass of the axion particle. We analyze data we collected from the Robert C. Byrd Green Bank Telescope in the L band and the Effelsberg 100-m Telescope in the L band and S band from a number of sources expected to produce bright signals of axion-photon conversion, including the Galactic center of the Milky Way and the nearby isolated neutron stars RX J0720.4-3125 and RX J0806.4-4123. We find no evidence for axion DM and are able to set constraints on the existence of axion DM in the highly motivated mass range between ∼5 and 11 μeV with the strongest constraints to date on axions in the ∼10-11 μeV range.