Hi, Taylor. This is actually a more complex question than it may at first seem. The reason is that there are a whole lotta assumptions and computations and probabilities involved, and the answer you get will depend upon where the data for the answer comes from. For example, AncestryDNA performs a form of phasing on DNA results that compares your results against population genotype models. MyHeritage is doing the same thing. They publish their own estimations of the benefits of using this method and claim improved accuracy in identifying small shared segments, but they don't publish the genotype databases they use, and there have been no peer-reviewed scientific research into the claims they make. Caveat emptor.
One study whose data I tend to trust was published in PLOS One. I know this table is probably going to get distorted, but let's give it a try.
Cousins # You Have # Detectable by DNA % Detectable
1 7.5 7.5 100%
2 38 38 100%
3 190 170.4 89.7%
4 940 431.5 45.9%
5 4,700 700.3 14.9%
6 23,000 943 4.1%
In brief, the study arrived at the number of cousins you would have by assuming two sets of grandparents and 2.5 total children per couple. That worked out to pretty well accommodate current population rates, and slower growth rates in some countries that had higher population rates in past generations. So it's a modestly conservative figure.
The number of those cousins that are detectable as matches by DNA is also somewhat conservative. But again, I personally have seen no peer-reviewed research that indicates a higher level of accuracy in detection. Claims of higher levels of detectability are made, but it's up to each of us to decide how much we can trust those claims at face value. A really quick example is this blog post.
After you enter that you've taken an autosomal DNA test on your own profile, WikiTree will, in overnight processing, look for other people who have also said they've taken an atDNA test and who connect to you through WikiTree's "paper-trail" of profiles. That search extends out to a maximum of eight relationship degrees: eight separate birth events with each of which the DNA gets recombined and commingled. That equates to 6g-grandparents up the tree, and 3rd cousins horizontally across the tree. From that chart above, you can see that there's about a 90% chance that 3rd cousins will share a detectable amount of DNA. But that percentage drops precipitously at the 4th cousin level; drops to under 50%. Personally, I understand why WikiTree had to draw the line somewhere, and 3rd cousins seems correct to me.
People who have recorded they've taken an autosomal DNA test will show up on the "DNA Connections" column at the right-hand side of profiles if they are no more distant to you, on "paper," than 3rd cousins. If the assumptions and info above are correct, Wikitree would display at most 236(ish) cousins (1st through 3rd), not counting parents/grandparents, siblings, aunts/uncles, etc. About 20 of those 236 would not share enough DNA with you to be detectable. Of course, we haven't been able to talk all our 3rd cousins into being DNA tested, so the actual number of cousins "linked" is going to be much lower than 236. :-)