Do you know the explanation why in modern Kohanim there is a paternal line encompassing about 50% of the Kohanim and then many lines encompassing the other 50% 1. Is it possible for a Kohen to be descended from a non-Kohen father in some cases?
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I suggest you look at the section of that Wikipedia article: Does a CMH prove Cohen ancestry?
It is all about probabilities, where you expect a more likely distribution, not a DNA test like in the movies where you prove an association to like 1 in a Trillion.
In any event, although it is called 'Y chromosomal Aaron' it could very well be some immediate descendant in any event (say Eliezar, his son and successor, or Pinchas his grandson, etc.) leaving a whole linage of Kohanim without the gene.
So although we know that in history there were fake Kohanim (I can't find a good source online for that, but I have definitely heard it in Shiurim), and any Kohain today has a presumption (Chazakah) of his status (also due to issues that could make him a Challal which don't affect the genetics discussion, other than the distribution among non-Kohanim), nevertheless the presence of the gene or lack thereof for a specific person is not conclusive one way or another, it is just a certain amount of evidence.
In some circles there is a story that only the "Rappaport" family has any "chazakah" of actually being Kohanim, and all other Kohanim are safek (in doubt) of actually being a Kohen or not.
As for the gene, it's in great dispute as to it's meaning. Further study needs to be done.