It appears that the hint and that my suggested 4th cousin appears to be actually a 5th cousin-once removed! Wow, that was easy.
I was ready to do that again and a few minutes later lightning did strike twice. In this instance my suggested 4th cousin appears to be a 3rd cousin-once removed.
I tried many other suggested matches linked to extensive pedigree charts but found no obvious matches.
Then I returned to the match I had previously found late last year and now the "Shared Ancestor Hint appeared. In this case the hint was technically correct but it showed the limits of this computer matching software. The DNA results predicted my match was a 4th cousin. The pedigree evaluation software matched us as 5th cousins. The DNA was correct in this case. The pedigree evaluation did not match "Debora LANHAM b. abt 1815 Kentucky" with "Deborah LANHAM ABT 1785 Virginia". This does not look like a technological failure to me. It looks like "pilot error" in the pedigree chart information to me. To mix my metaphors this is a small case of "garbage in garbage out."
So to summarize we have three predicted 4th cousins who appear actually to be:
a 5th cousin-once removed;
a 3rd cousin-once removed; and
a 4th cousin.
PS: Just as I was wrapping up this post, I noticed that there is another way to filter DNA matches "Has A Hint". I now have five more matches to investigate and they are predicted to be 5th or more distant cousins.