Limitations of y-DNA Testing
In general, y-DNA can be used for men to prove biological ancestry along the paternal line in a family. However, there are at least three significant pitfalls in using y-DNA to prove lineages for the family tree.
First, y-DNA analysis is hampered by the fact that y-DNA testing sites do not require any proof of a claimed ancestor. For example, on the Wyatt page at FamilyTreeDNA.com, there are currently 3 individuals who claim to be descended from the same well-known 16th century poet, Sir Thomas Wyatt, yet these individuals have very different y-DNA results. It is not possible that all 3 are descended along a paternal line from the poet: at most one is, and possibly none are.
Therefore, we cannot know, from y-DNA results alone, which individuals are actual descendants of their respective claimed ancestors. Only if the actual remains of the long-ago ancestor have been successfully y-DNA tested would it be possible to know which present day individual is the actual paternal-line descendant.
The second pitfall in y-DNA analysis is that a lineage might contain a father-son pair that is not biologically related. There are several reasons this can happen:
Infidelity (by the son's mother) is the most obvious. But another possible reason is adoption. In past centuries, mortality among young adults was high, and orphans might be adopted by close relatives. If an orphan boy were adopted by his mother's relatives, or by his father's sister's family, he would not share y-DNA with his adoptive father, yet he is related to the family (through other chromosomes).
Finally, there is a third, very uncommon circumstance that can cause the y-DNA to differ along the male line of descent: if a daughter marries a distant relative (or an unrelated man) with the same last name. For example, Eleanor Roosevelt married her 5th cousin once removed, Franklin Delano Roosevelt. Their sons would rightly assert a close relationship to Eleanor's uncle, Teddy Roosevelt. But y-DNA testing of the sons would not confirm that close relationship. The reason is that they got their y-DNA from the parent who was more distantly related to Teddy. If you tested the sons' other chromosomes, you would find many that matched Teddy's.
Thus it is possible that cousins who share a last name might not share the same y-DNA.
Considering again the 16th century poet, Sir Thomas Wyatt, we find that there were many Wyatt families in London in the 16th and 17th centuries, some related to Sir Thomas Wyatt, and some not. The children of marriage between a female descendant of Sir Thomas, and man from a different Wyatt family, would be true descendants of Sir Thomas Wyatt, and would carry the last name "Wyatt", but would not match Sir Thomas's y-DNA.
The lesson is that genetic analysis can supplement genealogical research, but it cannot replace old-fashioned historical, documentary research to prove our ancestral lineage.