You raise a great question: "We believe we are related, but DNA doesn't show connection". I will try to give three non-technical answers:
1. Your genetic (DNA) tree will be different than your "real" (genealogical) tree. You will always get DNA from parents, grandparents and even all 8 great-grandparents. Further back in time, you will likely get DNA from all 16 gg-gp's, but sometimes maybe only from 15 or 14. Each generation further back in time results in less "pieces" (cM's) of DNA being from all ancestors.
2. The DNA segments (cM) you receive will always be large enough so that you will share some DNA with a 2nd cousin. But, you could have a legitimate 3rd cousin that does not match any of your DNA. It could be that the segments you have from the common gg-gp's do not overlap.
3. When your DNA is created, there is something at the chromosome level known as crossover. The "non-sciencey" meaning is that chromosomes tend to recombine at specific break points. So a given chromosome will only have so many pieces and not more. For instance, if a chromosome only has 15 crossover sections, you will not get a segment from at least one of your 16 gg-gp's. It is sort of like musical chairs. In this example, 16 gg-gp's and only 15 crossovers, leaves at least one off. If two segments come from the same one gg-gp, than you will only have 14 gg-gp's represented in that chromosome. If you want to learn more about crossover, and how it differs by gender, see this blog posts by Roberta Estes: https://dna-explained.com/2017/11/09/concepts-dna-recombination-and-crossovers/ from 2017 and https://dna-explained.com/2019/09/17/crossovers-frequency-and-inheritance-statistics-male-versus-female-matters/ from 2019.
Wishing everyone stays safe and healthy,