The Collaborative International Dictionary of English v.0.48:

Possibly \Pos"si*bly\, adv.
   In a possible manner; by possible means; especially, by
   extreme, remote, or improbable intervention, change, or
   exercise of power; by a chance; perhaps; as, possibly he may
   recover.
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         Can we . . . possibly his love desert?   --Milton.
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         When possibly I can, I will return.      --Shak.
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The Collaborative International Dictionary of English v.0.48:

May \May\ (m[=a]), v. [imp. Might (m[imac]t)] [AS. pres. maeg
   I am able, pret. meahte, mihte; akin to D. mogen, G.
   m["o]gen, OHG. mugan, magan, Icel. mega, Goth. magan, Russ.
   moche. [root]103. Cf. Dismay, Main strength, Might. The
   old imp. mought is obsolete, except as a provincial word.]
   An auxiliary verb qualifying the meaning of another verb, by
   expressing:
   (a) Ability, competency, or possibility; -- now oftener
       expressed by can.
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             How may a man, said he, with idle speech,
             Be won to spoil the castle of his health!
                                                  --Spenser.
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             For what he [the king] may do is of two kinds; what
             he may do as just, and what he may do as possible.
                                                  --Bacon.
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             For of all sad words of tongue or pen
             The saddest are these: "It might have been."
                                                  --Whittier.
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   (b) Liberty; permission; allowance.
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             Thou mayst be no longer steward.     --Luke xvi. 2.
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   (c) Contingency or liability; possibility or probability.
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             Though what he learns he speaks, and may advance
             Some general maxims, or be right by chance. --Pope.
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   (d) Modesty, courtesy, or concession, or a desire to soften a
       question or remark.
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             How old may Phillis be, you ask.     --Prior.
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   (e) Desire or wish, as in prayer, imprecation, benediction,
       and the like. "May you live happily." --Dryden.
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   May be, & It may be, are used as equivalent to
      possibly, perhaps, maybe, by chance,
      peradventure. See 1st Maybe.
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WordNet (r) 3.0 (2006):

possibly
    adv 1: by chance; "perhaps she will call tomorrow"; "we may
           possibly run into them at the concert"; "it may
           peradventure be thought that there never was such a time"
           [syn: possibly, perchance, perhaps, maybe,
           mayhap, peradventure]
    2: to a degree possible of achievement or by possible means;
       "they can't possibly get here in time for the funeral?" [ant:
       impossibly]