Just as there are surnames which are very common, (such as Smith and Jones), and surnames which are uncommon, there are Haplotypes (a set of results from the Y-Chromosome that characterize you) with a high frequency of occurrence (aka common), and Haplotypes with a low frequency of occurrence (aka uncommon). The 12 Marker result from the Y-chromosome test is called a Haplotype, and can help determine if your DNA sample is common or uncommon.
When you compare a 12 Marker result to another 12 marker result of someone with the SAME surname, and the results match 12/12, there is a 99% probability that you two are related within the time frame included in the MRCA tables. If the match is 11/12, there's still a high probability that you are related IF the 11/12 match is within the same surname. If you compare a 25 Marker result to another 25 marker result for the SAME surname, and the results match 25/25, then there is also a 99% confidence that the two individuals are related…and at a much closer time interval then with the 12 marker test.
If you compare the 12 marker result to someone else who does not have the same surname, but the scores match, you are most likely NOT recently related. When we use the term recently related, we are talking about a time frame within the last 1000 years or 40 generations, a time depth that accommodates the earliest known use of surnames.
According to current theories, we are all related. The degree of relatedness depends on the time frame, or the number of generations between the participants and the common ancestor.
Dr. Luigi Lucca Cavalli-Sforza, Professor Emeritus, Stanford University, in his fascinating book: The Great Human Diasporas: The History of Diversity and Evolutions says that the total population of Europe was 60,000 people at the end of the last Ice Age, about 10,000 years ago. Now Europe has a population of 300 million people. This increase in population is almost entirely due to a natural increase in population rather then immigration from other continents. Keeping this in mind it is reasonable that many people alive today in Europe will match with other Europeans from BEFORE the time that our ancestors began the adoption of surnames, and when you match someone who has a different surname your first thought should be that the ‘connection’ is distant rather then recent.
The only exceptions to this are if an unannounced adoption or false paternity has taken place, but that is difficult to prove, although certainly not impossible.
If two 12 marker results match for two participants with the same surname, and the genealogy research shows a common ancestor in 1835, the DNA test has validated the research and proven that the two descendents are related. In this example, you have two items of evidence to support that the individuals tested are related…a documented paper trail and the DNA results. In addition, the research provided a precise time frame for the common ancestor.
Without the genealogy research, and where 2 participants with the same surname match on the 12 marker test, then the scientific answer to the degree of relatedness is that 50% of the time the common ancestor would have occurred within 14.5 generations, or within approximately 360 years. The range of generations for the common ancestor extends to 76.9 generations, or almost 2000 years for those cases where there is not a surname in common. Therefore the importance of a surname link is paramount to provide a comfortable conclusion of relatedness. Most of the time random matches with people with different surnames do not stand the test for extended DNA testing.