Bridgehead carbons generally have difficulties undergoing SN2 reactions because a bridgehead alkyl halide will be tertiary: bridgehead carbons are bonded to each of the two rings as well as the bridge between the two rings. Tertiary alkyl halides have a lot of difficulty undergoing SN2 reactions because the nucleophile would experience a lot of steric hindrance when trying to attack the alpha carbon. As a result, tertiary alkyl halides will generally undergo elimination reactions.
The greater the steric hindrance, the more difficult it will be for the nucleophile to attack.
A bridgehead carbon, however, will be especially unreactive to SN2 because of how SN2 proceeds via inverse of configuration. Inverting the stereochemical configuration of the bridgehead carbon will generate significant angle strain for the other carbons in the ring system, which especially for smaller cyclic compounds, will create a highly unstable product.