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)

Moby Thesaurus II by Grady Ward, 1.0:

32 Moby Thesaurus words for "residency":
   Hippocratic oath, abiding, abode, cohabitation, commorancy,
   domicile, dwelling, general practice, habitancy, habitation, home,
   house, inhabitancy, inhabitation, inhabiting, internship, living,
   lodging, medical practice, nesting, occupancy, occupation,
   practice of medicine, residence, residing, restricted practice,
   sojourning, squatting, staying, staying over, stopping, tenancy