If you look way down on this page, under "McKee Oral Traditions," there is some discussion. It is possible that G. W. McKee's genealogy, which mentions that 11 brothers came, may be responsible for spreading this.
In any case, there was a tradition that one descendant of Gavin (Guian) McKee (1750/1-1827), of Fleming Co., KY, had, that her ancestor was one of 11 brothers and 4 sisters (or something like that). Gavin was almost certainly a nephew of David McKee (c.1710-1795), father of the founder of McKeesport, who elsewhere is also claimed to be one of the 11 brothers. However, the earliest I can find him is probably in 1763 in Cumberland County, PA. Guian's possible father, however, was Robert McKee, who went to Chester Co., PA, by the early 1730s.
The oral tradition in my immediate family is that the McKees were seamen or something like that, and that they came to America on a Dutch ship. In Raymond McKee's *Book of McKee* there is a similar story of a McKee family in Ireland making their own boat and 10 or 11 brothers coming to America in it.
My interpretation of the 11 brothers story is that the brothers came together from Scotland to Ireland (maybe about 1685), not to America, and their descendants, including the Augusta/Rockbridge County McKees and the McKeesport McKees, and others, were descendants of this group. Of course that's just a theory.