You might want to look over this blog article: Genetic Genealogy and the Single Segment. It discusses a few points one might want to consider when one matches another person on just one segment.
One aspect touched upon there is how segment size changes as we match on more distant ancestors. It's true that we get on average half as much DNA on average from the typical ancestor in a generation immediately preceding a given generation of ancestors.
One might think this means the segment size inherited from that preceding generation is always smaller, but that is often not the case.
Inheritance of a stretch of DNA from a grandparent is predominantly an all or nothing situation; only occasionally does a stretch include genes from both grandparents instead of excluding all DNA from one grandparent or the other. For shorter stretches, it's close to a 50/50 chance whether you get any DNA at all from a particular grandparent within that stretch. (On the other hand, for long stretches like a whole chromosome it's quite likely you'll get some from each grandparent.)
The bottom line is that if you inherit a stretch of DNA from a distant ancestor at all, then the piece you inherit can be larger than you expect from the fact that the amount of DNA passed along in total halves with each generation.
In your case of a 28.6 cM segment shared with a sixth cousin, your common ancestor would have been a 5-great grandparent. Given that a segment is passed down, the likelihood of a segment being passed down one generation without being chopped down to shorter than 28.6 cM is about 72%. (This is based on the rule of thumb given in the linked article, but this "rule" begins to get too inaccurate above about 50 cM.)
Since there are 7 births between you and your 5-great grandparent, this would mean there is about a 10.2% chance that the segment you inherited from them is not reduced below 28 cM.
For you to share that segment with a sixth cousin, they also not only have to inherit a segment in that area, but it must be at least 28.6 cM in size. So for both of you to have the segment not shortened to a smaller size, your two segment copies each survived an equivalent gauntlent of inheritance at 10% each. Hence, there is about a 1% chance (10%*10%) of this size being retained by both of you.
Is 1% believable? To me, it's borderline: not ridiculous but unlikely. So I'd think there could be an error somewhere, but know that there may not be.
Sources of error are not likely to be in the DNA files or GEDmatch matching in my estimation. To me it seems more likely that there is some aspect of the two family trees that is unknown or incorrect. For instance, a spurious extra generation might occur in one of the trees which together show the sixth cousin relationship. Another possibility is that there is another unknown relationship among the two sets of ancestors, one that is a bit closer than the documented one and which actually is the source of the shared segment.