The Collaborative International Dictionary of English v.0.48:

Incompatibility \In`com*pat`i*bil"i*ty\, n.; pl. -ties. [Cf.
   F. incompatibilit['e].]
   The quality or state of being incompatible; inconsistency;
   irreconcilableness.
   [1913 Webster]

WordNet (r) 3.0 (2006):

incompatibility
    n 1: the relation between propositions that cannot both be true
         at the same time [syn: incompatibility, mutual
         exclusiveness, inconsistency, repugnance]
    2: (immunology) the degree to which the body's immune system
       will try to reject foreign material (as transfused blood or
       transplanted tissue)
    3: the quality of being unable to exist or work in congenial
       combination [ant: compatibility]

Moby Thesaurus II by Grady Ward, 1.0:

114 Moby Thesaurus words for "incompatibility":
   Discordia, Eris, ambiguity, ambivalence, antinomy, asymmetry,
   autism, bashfulness, chill, chilliness, clash, clashing, coldness,
   conflict, contention, contrariety, contrast, coolness, departure,
   deviation, difference, disaccord, disaccordance, disaffinity,
   disagreement, disconformity, discongruity, discord, discordance,
   discordancy, discrepancy, discreteness, disharmony, disparity,
   disproportion, disproportionateness, dissent, dissimilarity,
   dissociability, dissonance, distinction, distinctness, divergence,
   divergency, diversity, enmity, equivocality, far cry, friction,
   frost, heresy, heterodoxy, heterogeneity, iciness, inaccordance,
   incoherence, incommensurability, incompatibleness, incongruity,
   inconsistency, inconsonance, inequality, inharmoniousness,
   inharmony, inhospitality, inimicality, insociability,
   irreconcilability, jangle, jar, mischief, mixture, mopishness,
   moroseness, nonconformability, nonconformity, noncooperation, odds,
   open conflict, opposition, otherness, oxymoron, paradox,
   personal conflict, rub, self-containment, self-contradiction,
   self-sufficiency, separateness, social incompatibility, strain,
   strained relations, sullenness, tension, unamiability,
   uncommunicativeness, uncompanionability, unconformability,
   unconformity, uncongeniality, uncordiality, unfriendliness,
   ungeniality, ungregariousness, unharmoniousness, unlikeness,
   unorthodoxy, unpleasantness, unsociability, unsociableness,
   variance, variation, variegation, variety



Bouvier's Law Dictionary, Revised 6th Ed (1856):

INCOMPATIBILITY. offices, rights. This term is used to show that two or more
things ought not to exist at the same time in the same person; for example,
a man cannot at the same time be landlord and tenant of the same land; heir
and devise of the same thing; trustee and cestui que trust of the same
property.
     2. There are offices which are incompatible with each other by
constitutional provision; the vice-president of tho United States cannot act
as such when filling the office of president; Const. art. 1, s. 3, n. 5; and
by the same instrument, art. 1, s. 6, n. 2, it is directed that "no senator
or representative shall, during the time for which he was elected, be
appointed to any civil office under the authority of the United States,
which shall have been created or the emoluments whereof shall have been
increased, during such time; and no person holding any office under the
United States, shall be a member of either house, during his continuance in
office."
     3. Provisions rendering offices incompatible are to be found in most of
the, constitutions of the states, and in some of their laws. In
Pennsylvania, the acts of the 12th of February, 1802, 3 Smith's Laws of Pa.
485; and 6th of March, 1812, 5 Sm. L. Pa. 309, contain various provisions,
making certain offices incompatible, with each other. At common law, offices
subordinate and interfering with each other have been considered
incompatible; for example, a man cannot be at once a judge and prothonotary
or clerk of the same court. 4 Inst. 100. Vide 4 S. & R. 277; 17 S. & R. 219;
and the article Office.



The Devil's Dictionary (1881-1906):

INCOMPATIBILITY, n.  In matrimony a similarity of tastes, particularly
the taste for domination.  Incompatibility may, however, consist of a
meek-eyed matron living just around the corner.  It has even been
known to wear a moustache.