"any event which has caused a break in the link between an hereditary surname and the Y-chromosome resulting in a son using a different surname from that of his biological father" (Source)When Selig and Julius, the sons of my ancestor, Samuel Dudelczak, immigrated to America and changed their surnames to Feinstein and Odelsohn, these were Non-Paternity Events. But that's not usually what is thought of when one hears the term.
In her Records of the Van Every Family, 1947, Mary Blackadar Piersol theorized that McGregory Van Every might be the son of Martin Van Iveren and an unidentified McGregory. Later researchers have wondered if it is equally possible that he could have been the son of a McGregory male and an unidentified Van Every female.
First, here is a transcription of relevant passages from Piersol's work indicating her sources (and lack thereof).
No entry has been found of the names of McGregory's parents due, probably, to gaps in the records of several of the churches of the Hudson Vallley. Indirect evidence, however suggests that his father was Martin Van Every and his mother possibly a member of the MacGregory family. His first children were named Francis and Patrick, the former a name from the Jaycocks family, the latter a favourite in that of the Colonel. Moreover with McGregory a significant change occurs in Van Every names. Previously they had all been Dutch but his is of Scotch derivation and among his sons we find such Scotch names as David, William, Peter and Andrew. Moreover one line of the Van Every family in Canada holds strongly the tradition that it has in its ancestry "a Scottish officer by the name of MacGregory." The wording is that of Anne Laurence Van Every, great-granddaughter of McGregory Van Every. (Piersol, p.98)
There is evidence of intimate acquaintance in later years between the descendants of the Colonel and those of Burger Van Iveren … Not only did they live in the same general locality, Ulster and Orange Counties, NY., but their names appear in the records of the same churches and are linked by the same documents. When the Colonel’s son, Patrick, made his will, Francis Jaycocks signed as witness. The records of the Zion Lutheran Church at Lunenburg, now Athens, show that the Colonel’s grandson, also named Patrick, witnessed the baptism of Maria Jaycocks, and the banns for her marriage to McGregory Van Every were recorded by the Dutch Reformed Church at Poughkeepsie, N.Y. (Piersol, p. 97-98)
As there is at present no documentary evidence as to the exact relationship I should hesitate to affirm it. But, as it was common knowledge of my Mother, and her Father, I would equally hesitate to deny it. The widow of Wm. Van Every lived with her son Joseph until her death, and my Mother, born in 1839, remembered her well. The legend is not without foundation as Dr. Piersol’s research shows. There was intimacy more than usual, which strongly suggests relationship, such as witnessing the will of the Colonel’s son, etc. The period to which it relates is not so remote as to be beyond the age of credible tradition. – J.J. Charteris-Thomson (Piersol, p.98-99)As can be seen, her conclusions weren't based on solid documents. I have had my DNA tested at both Ancestry and FamilyTreeDNA. Of all matches that have Van Every ancestors in their known tree (that is, a tree they have made available), all are descendants of David Van Every and Sarah Showers, David being one of McGregory Van Every's sons. Cousins of my generation, with whom David and Sarah are my most recent common ancestors, are my fifth cousins. Both websites indicate 5th-8th degree cousins. There are matches that I have confirmed are 2nd, 3rd, 4th and 5th cousins through my Van Every lines. And then it stops.
One could attribute it to the difficulty of tracing one's lineage that far back. David Van Every served in the American Revolution. Except, Piersol's work has been available since 1947 to genealogists researching the surname, and her research documented several other branches.
The absence of a 6th cousin Van Every match, or greater, doesn't mean one doesn't exist. And it certainly doesn't prove an NPE. Perhaps I don't have enough DNA from McGregory. Even if there is an NPE, since I don't have any matches with descendants of David's siblings, the NPE could easily be with David and not McGregory. (Or both.)
Unless another descendant of David and Sarah [Showers] Van Every has had a different experience with DNA cousin-matching, the only way to resolve the question is for a group of Van Every males of varying descent to take the Y Surname test. I have no ability to make this happen beyond bringing the issue up.