Absolutely they can...if you use all lines of evidence. A family that is Jewish, will the name Kohen with the DNA markers that all other Jews named Kohen share...
Personally I don't doubt that there are valid descendants of Aaron etc.
However, technically what you are proposing is a circular argument. In other words, These people claim to have Aaron's DNA and we know that it is Aaron's DNA because these people have it.
All your argument technically proves is that the people in question had a common ancestor. It doesn't tell you who that common ancestor was. At best this could be seen as a rather small piece of evidence that might support their tradition that they are Kohanim.
In order to actually prove descent from Aaron, you'd have to have Aaron's DNA in some indisputable form. We don't have that.
However, I've seen a lot of people comment about the Ashkenazi etc, and there is a common theory that many of the European Jews are not racially Jewish but are descended from gentiles who converted to Judaism. I believe that this idea has been largely disproved by genetic testing. As far as I've been able to see most of the Ashkenazi have been shown to have Israelite DNA going back to the middle east, and are not of purely European/Caucasian race.
To a certain extent that question is moot, however, because Jewishness and Israeliteness is biblically not racially defined but religiously defined. Israelites who forsook the Covenant cut off from Israel and were no longer part of the people. Likewise gentiles could join the covenant and become part of Israel.
As a final note, this is pure speculation, but it is statistically probable that most people in the world have some link to Israelite descent. Do to the nature of the way ancestry spreads out as you go down through time, and the fact that the 10 tribes were scattered among the nations, and the Jews were also scattered among the nations, combined with how long ago this happened, it is probable that virtually everyone in the world could have some Israelite ancestry.
For example, it is statistically estimated that virtually every person in Europe today can claim descent from Charlemagne. Similarly in Ireland virtually every person can claim descent from King Niall. Anyone in history who had many children is likely to be an ancestor to much of the world's population today.
The reason for this is that as you go back in time the number of your ancestors increase exponentially. At one generation you have your parents (2). At to Generations you have your grandparents (4), then your greats (8) then your great-greats (16). By the time you go back a thousand year, the number of your ancestors is greater than the population of the world.
This means 2 things. First, there is necessarily a lot of intermarriage within extended families. Second, it means that every person from the ancient past, who's line didn't die out, is probably an ancestor to most living people today.
The exception to this would be that regions like China and Europe which had very little contact obviously couldn't have much intermarriage so if you are European, you are probably related to every person in medieval Europe, but not to every person in medieval China.