Previous work on the emission from isolated old neutron stars (IONSs) accreting from the interstellar medium (ISM) has focused on gravitational capture-i.e., Bondi accretion. We propose a new class of sources that accrete via magnetic interaction with the ISM. While for the Bondi mechanism the accretion rate (M)over dot(Bondi) decreases with increasing neutron star velocity, in magnetic accretors (MAGACs) (M)over dot(MAGAC) increases with increasing neutron star velocity ((M)over dot(Bondi) proportional to (-3) vs. (M)over dot(MAGAC) proportional to v(1/3)). MAGACs will be produced among high-velocity (greater than or similar to 100 km s(-1)), high magnetic field (B > 10(14) G) radio pulsars-the "magnetars"-after they have evolved first through magnetic dipole spin-down, followed by a "propeller" phase (during which the object sheds angular momentum on a timescale less than or similar to 10(10) yr). The properties of MAGACs may be summarized thus : dipole magnetic fields of B greater than or similar to 10(14) G; minimum velocities relative to the ISM of 25-100 km s(-1) or higher, depending on B, well below the median in the observed radio pulsar population; spin periods of greater than days to years; accretion luminosities of 10(28)-10(31) ergs s(-1); and effective temperatures kT(eff) = 0.3-2.5 keV if they accrete onto the magnetic polar cap. We find no examples of MAGACs among previously observed source classes (anomalous X-ray pulsars, soft gamma-ray repeaters, or known IONs). However, MAGACs may be more prevalent in flux-limited X-ray catalogs than their gravitationally accreting counterparts.