Hi, Jonathan. I think I understand the end result you're asking about, and my opinion would be a qualified, "No; there generally cannot be a concise and accurate statement made, based on DNA, that 'these two people can't be related.'"
But as G2G's most notorious EWCS (Excessive Word-count Scofflaw), of course there are qualifications! And explanations! And examples! <We pause now for a brief moment of sonorous groaning>
Before I get into it, first I want to note that we buried and blessed the "confirm" etymology and definition controversy a few years ago. Ain't gonna bring that up again, as such. So this isn't about that. I'll try not to use the word "confirm" unless I'm specifically addressing it as a WikiTree "Confirmed with DNA" policy or guideline. Suffice to say in that context it does not mean "establish the truth or correctness of."
Second, DNA is still very much a growing and evolving science. For example, the version of the human genome map that all the major testing and reporting companies currently use for autosomal DNA was superseded in 2013. So not only is the current reference genome different, but what we've learned about DNA in just those intervening eight years--from technology to techniques to application--has been enough to, quite literally, fill multiple books.
In terms of genealogy, DNA is very, very seldom binary: not a yes/no, either/or. Some genealogists think it is--or at least want it to be--but it almost never is, relatively speaking (pun intended). So it requires evidence analysis just like any other potential piece of evidence...though I've offered the opinion before that DNA introduced a type of evidence that was new to genealogy and that required a different type of knowledgebase and skillset to effectively perform.
I'll rephrase your question a bit, just to make it a little more specific for my own purposes: "Can we verify that these two test takers, based on a given set of actual DNA results, are not related in the genealogical timeframe?"
As Judy Russell has written, "Negative evidence is the hardest type of evidence to understand or use in genealogical research. By definition, [it is] a 'type of evidence arising from an absence of a situation or information in extant records where that information might otherwise be expected...'"
That "might otherwise be expected" part is key. An example of simple deductive use of negative evidence that both Judy and Elizabth Shown Mills have used is the Arthur Conan Doyle short story, "Silver Blaze." In it, a pivotal moment comes when Holmes realizes that a guard dog hadn't barked an alert at the scene of the crime:
"Is there any point to which you would wish to draw my attention?"
"To the curious incident of the dog in the night-time."
"The dog did nothing in the night-time."
"That was the curious incident," remarked Sherlock Holmes.
In the lab, most of what's done with DNA is inductive, based on testing and observation...though things like imputation can crossover to deduction. With deductive reasoning we're making an inference based on expectations or what we believe are widely accepted facts. Negative evidence--which is a perfectly valid form of evidence--is always deductive. We can't actually measure something that isn't there...just ask Schrödinger's cat. Ergo the old trope: absence of evidence is not evidence of absence.
Mostly all types of evidence, either positive or negative, live on a sliding scale from "meaningless" to "wow, super-important." To make that a bit more meaningful, call it from 0 to 1, and all the fractions in between. Not meaning to drive things down a rabbit hole, but all the items of evidence can interact with each other. The evaluation of the weight of a piece of evidence is dynamic: its importance is relative to all the other bits of evidence available, and can change with the introduction or removal of a different piece (or pieces) of evidence.
Simple genealogy example. John Smith's will left a tract of land to his son, James Smith. If that's the only document we have, we rate it as a 1, "wow, super-important," and record James as John's son. Then a couple of years later we discover custodial papers from a county court that show John Smith adopting a James Jones, age 6, parents deceased. Very much changes the evidence weighting of John's will.
Still percolating on two of my lonely, remaining glial cells is the notion of adapting Bayesian reasoning and the statistical concept of "likelihood ratio" to assign dynamic, quantitative values to evidence weighting in genealogy. To describe levels of confidence in traditional genealogy, terms like "apparently," "possibly," and "probably" are used, and especially as DNA enters the picture those become a bit too ambiguous to be very useful, IMHO.
"Negating evidence" is a legal term (considering tax day in the U.S. is next Monday, it might be worth knowing that the IRS uses the term in its Internal Revenue Manual, Part 9, "Criminal Investigation," Chapter 5, Section 9; ahem). But in the physical and life sciences--just like there is no ultimate "1" weighting designation because "when it comes to science, proving anything is an impossibility" (Ethan Siegel, writing for Forbes)--there's no "0" for impossible, either.
Now, I've used "negating evidence" as distinguished from "negative evidence" when it come to genetic genealogy, but infrequently...and even then it's more like a value of 0.001 rather than 0. Heck, even actual forensic testing is almost never absolute; there are still deduction and preponderance of evidence considerations involved, despite what we see in reruns of Law and Order. An example of something that can put the kibosh on a full "0" with DNA is an an allogeneic bone marrow transplant. There are ways to get around it in the lab, but our little microarray tests don't take those measures and a parent who has received bone marrow and T cells from another person may have an AncestryDNA report that shows, incorrectly, they aren't biologically related to the tested children.
In terms of "negating evidence," I reserve that for mismatches in high-level haplogroups. If two people take mtDNA tests and we see haplogroups of H and U, then we can negate the hypothesis that they share a matrilineal line ancestor in the genealogical timeframe. Likewise if one man tests as a R1b Y-haplogroup, and another is G: no patrilineal line ancestor in the genealogical timeframe.
But we have to be careful even with haplogroups when it comes to evaluating a common ancestor. Here's an ongoing, real-world example from one of my FTDNA Group Projects. Persons A, B, and C are thought, on paper, to descend from three different sons of the same man who was born circa 1650. Person A takes the Big Y test and the results show R1b with the following (partial) SNP-positive hierarchy: R-L151 > U106 > Z381 > Z301 > S1688 > FGC51534. Person B takes a Big Y test and matches A on all of those SNPs with three private variants; A and B are an STR genetic distance of 3 at 111 markers, 1 at 37 markers.
Person C then tests a single SNP, R-Z301. It tests negative. YFull estimates TMRCA for Z301 at 4,600 YBP. So done deal, right? C is not patrilineally related to A and B. Negating evidence.
But wait... C also took a 37-marker STR test. At that level he is a GD of 3 with B, and 4 with A. TiP report says over a 90% likelihood of CA at 12 generations. Note that this level of matching is perfectly acceptable per WikiTree guidelines to mark all three men as "Confirmed with DNA" back to that 1650 ancestor. And at 37 markers, C also has GD 2 and 3 matches with 5 other men who are also STR matches at that level to A and B. In fact, all of C's 37-marker matches are shared among A, B, and C.
Unfortunately, C passed away and we can't order another upgrade against his stored sample. But is it possible he could still be related to A and B just as the paper trail says? That the lab test for the single Z301 SNP was in error, or that C has a back-mutation at Z301 and that all the other SNPs would test positive? We don't know, but the possibility absolutely exists.
On the autosomal DNA side, the most recent data comes from Amy Williams at Cornell University. The most distant cousin that is expected to share DNA with you 100% of the time is 1C1R (also equivalent to a half 1st cousin). A 2nd cousin should share DNA with you 99.98% of the time...but that doesn't quite reach 100%. A half 1st cousin 1x removed, 99.97%; a 2C1R, 99.1%. So genealogically speaking, the farthest you can go with an Accuracy = almost "1" for negative evidence is the grandparents; fairly close to "1" is great-grandparents. However, we can't actually get to "1" because, again, we can never be 100% certain of the laboratory accuracy.
It isn't as difficult for positive evidence. If you share a couple of reasonably-sized segments for a total of, say, about 25cM, you're almost certainly related. Probably around 3C1R or 4C. Lab error isn't as much a factor because to arrive at that there have been a couple of thousand unique SNPs tested that matched along those segments.
Negative evidence needs to be evaluated much more stringently, and the constraints to those evaluations don't allow a great deal of leeway. In no small part because for genealogy the paper trail always has to drive the hypotheses. Back to the example of John Smith's will and the discovery that son James was adopted: what if five of James's descendants had been yDNA tested--thus seeming to show a consistent preponderance of evidence--and a sole descendant of John's other son Abraham had tested. But that descendant of Abraham is a different high-level haplogroup from the other five test-takers. If the adoption record had never been discovered, we might treat the test results of Abraham's descendant as negating evidence...when in fact he was the only one of John's biological descendants ever tested.