No, the paper trails are not in error, and the DNA does not show that.
Y-DNA passes virtually unchanged from father to son except for random "mutations" which occasionally happen.You didn't state how many generations had passed from the first two to their common ancestor, or their relation to each other. That they match 67/67 just means that there have been no mutations in either line on those 67 markers from them back to their common ancestor.
Now you have person three, whose ancestor is a first cousin to the common ancestor of the first two. That he matches them 67/67 just means there have been no mutations on those markers back to the grandfather of the common ancestor of the first two, which is the first generation all three have in common.
While Y-DNA cannot prove who the common ancestor of men that match is, the fact that they match each other 67/67 shows they do have a common direct line male ancestor, which would be evidence to support the paper trail.The paper trail still has to be evaluated at each step, as it is possible that a researcher could have the wrong person (brother or cousin) for the correct father in one generation, but because they would all share the same y-DNA their descendants would closely (or exactly) match.