The answer is probably no. Most of the black holes that we know about have masses above 5 solar masses, whereas all of the neutron stars have mass below 2.5 solar masses. This gap suggests that just adding mass to neutron stars is not a common way to form black holes. See the summary plot below from Ozel et al. (2012).
If black hole formation by the simple addition of more mass to a neutron star was common, then we might expect to see many more objects populating the mass range just above the maximum supportable mass of a neutron star - which is thought to be around 2.2-2.5 solar masses, but almost certainly lower than about 3 solar masses.
What black holes and neutron stars share is progenitors that are massive stars. One school of thought is that there is a bifurcation at the ends of their lives, during the supernova core collapse that ends up either with a black hole above 5 solar masses or a neutron star of much lower mass. However, another school of thought is that although possibly rare, some low mass black holes, formed when neutron stars accrete more mass, are out there, but very difficult to spot observationally.