My great-grandfather, Yankel Feldman of Skapiskis, Lithuania was the brother
of S's great-great-grandmother. We are therefore 3rd cousins once removed.
However a comparison in Gedmatch tells us: " No shared DNA segments found".

I am a layman trying to understand this statement and will be grateful to
the mavens if they put me on the right track.

I understand it as follows:
The DNA tests are for a certain number of genetic markers and not for the
entire human genome therefore the statistical reliability of the tests
should drop. However how much do they drop?

Siblings have on the average 50% DNA in common
First cousins 25%
2nd cousins 12.5%
3rd cousins 6.25%
4th cousins 3.125%
Since S and I are between 3rd and 4th cousins we should expect 4.6875% .
How does that translates into cM?

Now theoretically a brother and sister could have no DNA in common if one
sibling received one set of chromosomes and the other the completely
different set of each parents. Since there are crossovers between
chromosomes that is probably impossible but if it is possible it would
probably be one chance in billions.
When we talk of average we have to relate to deviations >from average eg in
the case of siblings what percentage of siblings would have an overlap of
between 25% - 75% and what percentages above and below these limits?

My family connection with S is backed up by a fair body of family narrative
and documentation. What is the statistical probability of us not having DNA
segments in common?

Jules Feldman
Kibbutz Yizreel