The Collaborative International Dictionary of English v.0.48:

Residency \Res"i*den*cy\ (-den-s?), n.
   1. Residence. [Obsoles.]
      [1913 Webster]

   2. A political agency at a native court in British India,
      held by an officer styled the Resident; also, a Dutch
      commercial colony or province in the East Indies.
      [1913 Webster]

WordNet (r) 3.0 (2006):

residency
    n 1: the act of dwelling in a place [syn: residency,
         residence, abidance]
    2: the position of physician who is receiving special training
       in a hospital (usually after completing an internship)