That is exactly the challenge. Genetic genealogy is a research-intensive, collaborative effort. My best matches have been ones in which I exchanged information with the match, we both did supplementary research, and then figured out the connection together. Sometimes that required a correction in one or both of the family trees.
You can also come up with surprises. I have a well-documented 4th cousin relationship on paper with a man who shares my surname. But DNA testing revealed that we are not genetically related at alll and that there is a non-paternal event (adoption, surname change or out of wedlock birth) somewhere in his paternal lineage.
In terms of a breakthrough, not long ago I worked with one of my close matches and together we were able to confirm that person's suspicion that there was a non-paternal event in her family involving one of my ancestors. Without DNA testing we never would have been able to verify what had been a family rumor. There are a number of stories on the various message boards, here and with other companies, about adoptees or children of donors who were able to combine DNA testing with diligent research and discover their families of origin.