Recent observational results for the masses and radii of some neutron stars are in contrast with typical observations and theoretical predictions for "normal" neutron stars. We propose that their unusual properties can be interpreted as the signature of a dark matter core inside them. This interpretation requires that the dark matter is made of some form of stable, long-living or in general non-annihilating particles, that can accumulate in the star (Sandin and Ciarcelluti (2009). In the proposed scenario all mass-radius measurements can be explained with one nuclear matter equation of state and a dark core of varying relative size. This hypothesis will be challenged by forthcoming observations and could eventually be a useful tool for the determination of dark matter.