Antilogy

According to answers.com, citing the Columbia Encyclopedia, obbligato is a musical term embodying a contradiction (an antilogy):

(ŏbləgä'tō) [Ital.,=obligatory], in music, originally a term by which a composer indicated that a certain part was indispensable to the music. Obbligato was thus the direct opposite to ad libitum [Lat.,=at will], which indicated that the part so marked was unessential and might be omitted. Misunderstanding of the term obbligato, however, resulted in a reversal of its meaning; when a violin part, for example, is added to a song it is called a violin obbligato, whereas it may be a superfluous ornament for which ad libitum would be a more precise direction.

In other words, obbligato can mean a part is either essential (indispensable) or superfluous (optional). But according to another source, obbligato is a classical musical term for countermelody:

In a piece whose texture consists clearly of a melody with accompaniment (i.e., a homophonic texture): a countermelody is an accompanying part with distinct, though subordinate, melodic interest. If the melodic interest were not subordinate, the texture would be polyphonic: two or more melodies of more or less equal melodic importance.

A notable instance of obbligato:
Procol Harum – A Whiter Shade of Pale (Matthew Fisher, Hammond organ obbligato)

A notable instance of melody/countermelody:
Billie Holiday (vocal, melody), Lester Young (tenor sax, countermelody) – A Sailboat in the Moonlight

A notable instance of polyphony:
Derek and the Dominos – Layla (Eric Clapton and Duane Allman, guitars)