In the case of ordinary accretion, neutron stars tend to expell most accreted matter. Due to their intense magnetic fields, this occurs mainly at the poles. When the poles of neutron stars lay close enough to our line of sight, it produces a pulsar. Pulsars invariable appear to 'spin down', signifying a slowing of rotation. If a neuton star were rapidly gaining mass, it should tend to rotate faster. In the case of inspiralling binary neutron stars, mergers are thought to account for short gamma ray bursts, consistent with your idea. Formation of a black hole is probably inevitable in these cases. About 5% of neutron stars are observed to have a neutron star companion.