The Collaborative International Dictionary of English v.0.48:

Competence \Com"pe*tence\, Competency \Com"pe*ten*cy\, n. [Cf.
   F. comp['e]tence, from L. competentia agreement.]
   1. The state of being competent; fitness; ability; adequacy;
      power.
      [1913 Webster]

            The loan demonstrates, in regard to instrumental
            resources, the competency of this kingdom to the
            assertion of the common cause.        --Burke.
      [1913 Webster]

            To make them act zealously is not in the competence
            of law.                               --Burke.
      [1913 Webster]

   2. Property or means sufficient for the necessaries and
      conveniences of life; sufficiency without excess.
      [1913 Webster]

            Reason's whole pleasure, all the joys of sense,
            Lie in three words -- health, peace, and competence.
                                                  --Pope.
      [1913 Webster]

            Superfluity comes sooner by white hairs, but
            competency lives longer.              --Shak.
      [1913 Webster]

   3. (Law)
      (a) Legal capacity or qualifications; fitness; as, the
          competency of a witness or of a evidence.
      (b) Right or authority; legal power or capacity to take
          cognizance of a cause; as, the competence of a judge
          or court. --Kent.
          [1913 Webster]

   5. the quality of being adequately or well qualified
      physically and intellectually, especially possession of
      the skill and knowledge required (for a task).
      [WordNet 1.5 +PJC]

WordNet (r) 3.0 (2006):

competence
    n 1: the quality of being adequately or well qualified
         physically and intellectually [syn: competence,
         competency] [ant: incompetence, incompetency]

Moby Thesaurus II by Grady Ward, 1.0:

128 Moby Thesaurus words for "competence":
   ability, ableness, absolute power, absolutism, address, adeptness,
   adequacy, adequate supply, adequateness, adroitness, airmanship,
   appropriateness, artfulness, artisanship, artistry, authority,
   bare minimum, bare sufficiency, bravura, brilliance, caliber,
   capability, capableness, capacity, cleverness, command, competency,
   condition, constituted authority, control, coordination, craft,
   craftsmanship, cunning, deftness, delegated authority, dexterity,
   dexterousness, dextrousness, diplomacy, divine right, efficacy,
   efficiency, enough, exact measure, expertise, facility, faculty,
   finesse, fitness, fittedness, flair, genius, grace, grip,
   handiness, horsemanship, indirect authority, ingeniousness,
   ingenuity, inherent authority, jus divinum, just enough, know-how,
   lawful authority, legal authority, legitimacy, marksmanship,
   mastership, mastery, maturity, might, minimum, power,
   practical ability, preparedness, prerogative, proficiency, prowess,
   qualification, quickness, readiness, regality, resource,
   resourcefulness, right, right amount, rightful authority, ripeness,
   royal prerogative, satisfaction, satisfactoriness,
   satisfactory amount, savoir-faire, savvy, seamanship, seasoning,
   skill, skillfulness, style, sufficiency, sufficient,
   sufficientness, suitability, suitableness, suitedness,
   susceptibility, tact, tactfulness, talent, technical brilliance,
   technical mastery, technical skill, technique, tempering,
   the goods, the say, the say-so, the stuff, timing, trim,
   vested authority, vicarious authority, virtuosity, what it takes,
   wit, wizardry, workmanship