Initial draft of an article.  To be expanded on with diagrams given to elaborate and visualize the information summarized above.  Check back for improvements.

There are a lot of aspects to Genetic Genealogy that yield inconclusive results due to variation.  Some of this variation is due to the testing technique and how results are reported.  For example, the testing technique cannot determine which chromosome copy a particular SNP is found on for autosomal chromosomes.  The test simply reports the value from both copies. But much of the variation is introduced by the biological processes itself.  Like the recombination (or cross-overs) that cause matching segments to get smaller with each generation and dilute your Autosomal DNA matching with relatives fairly quicly as the number of generations separating you increases.  Combined with the sampling variation of only getting half the chromosomes from each parent, it is a wonder we can make any conclusions at all.  As such, it takes analysis along with what is known about the people involved to extract useful information from genetic testing in most cases.  

We liken the process of using genetic results to that of the fall pruning of your pedigree tree.  The more analysis you do, and the more information you have available, the more pruning you can do till you are left with only one possible solution (or at least, a very narrowed down one).  As a result, the more distant, known relatives you can have tested, the better a job at pruning you can do.  But not knowing your pedigree tree does not give you much of a framework from which to prune.  And thus DNA testing on its own is not helpful. Maybe another analogy will help.

Trying to draw conclusions from genetic testing in the absence of genealogical knowledge is like trying to do a jigsaw puzzle with the grey-backs facing upwards.  You are trying to match and form the puzzle purely from the geometric shape.  And as we all know, such puzzles have many similar or identical shaped pieces.  So without having the genealogy, or visual clue of the photo image on the front, your likelihood of coming up with the right solution is very limited.  Genetic Genealogy cannot be conducted in a vacuum or independent of good, traditional genealogical work.  You need to see the printed image on the front of the pieces.

With that said, genetic testing can be used to help pose and resolve tough questions that traditional genealogy is not helping resolve.  The obvious case is adoptions that were not recorded or really any unknown parentage that is in question. But also, those brick walls you have. For example, if you have a great-grandmother who you simply know nothing about before her marriage, and no record research has been forthcoming.  No maiden name is known, no siblings, and no place of origin.  What is different from this an adoption?  In cases like these, get known but more distant realtives tested.  Especially those descended from that great-grandmother along lines different than your own.  And from the great-grandfather’s siblings and his descendants.  In this way, when you get a new autosomal match, you can quickly prune the possible tree branches based on who is matching and who is not. And maybe the person you match with has enough genealogical information known to at least give you a small gap to bridge in finding that elusive great-grandmother.

Then, looking at the strengths of the matching, you can get a rough range of the generation count from each matching relative.  This gives you rings from each relative of a certain genealogical distance away based on the strength of the match.  By analogy, when you place the relatives on your pedigree tree, you get an intersection point (or rough area of overlap) of these rings.  This is exactly how GPS triangulation works to pinpoint your location on earth. Albeit with the caveat we are looking at “locations” in time on a pedigree chart and not in physical space.  But the analogy holds.  Additional relatives tested is like having additional GPS satellites available that contribute more information. The more relatives you get tested that are off from the more branch points on your pedigree tree, then the more chance you have of getting an exact pinpoint of where that match fits. The GPS analogy holds as the more GPS satellites from the more varied locations in space, the better the triangulation to a specific location in space.  When matching with someone in autosomal SNP testing, If neither of you have genealogical knowledge of your own tree, and neither of you have many relatives tested, then the likelihood of the genetic testing helping is very small to non-existent.