OK--I've now gone through every one of my more than 3000 matches to identify all of the people with ancestors from Northern Ireland. The spreadsheet ran to over 2800 lines. I have found more than 225 instances in which I match with two or more individuals who have the same ancestor in their trees. There were 26 cases in which I matched with three such individuals, 6 with four matching individuals and one with six.
I decided to concentrate on the cases with the greatest number of matches to a common ancestor. They confirm, almost certainly, that my maternal grandmother had a large number of ancestors who were part of the early 18th century Scots-Irish immigration to America. In particular, there is one family from County Antrim with her surname. Although inconsistencies and gaps in the records prevent a firm conclusion, this location is consistent with birth records that I found for members of her family.
In the case with six matching individuals that family was from the home town of my paternal grandmother's family, confirming that they too had ancestors who were part of the earlier migration.
I made a similar, but smaller, list of my matches with ancestors from County Cork, home of my father's family. There were about two dozen shared ancestors. I couldn't draw any firm conclusions, but what I found was that there were shared matches with some prominent families from the 17th century, and merchant families in the 18th. This is tenuous evidence that is consistent with my theory that my father's family gradually transformed from landowners to tenant farmers over the course of two centuries.
Altogether this research took me about a month of work, a couple of hours per day. It would have been much simpler with adequate search tools. Also, without the ability to compare matching segments the conclusions are necessarily speculative.
Nevertheless, it has been a valuable exercise and has given me new things to ponder about my family.