Can any kohanim alive today trace their lineage all the way back to the High Priest Aaron? If not, who can go the farthest?
Although I personally do not know of someone with such links, I don't doubt the possiblity that one could be produced. Geneology is my hobby, and although Jewish geneology is very difficult, for a variety of reasons, going far back 2000 years is not impossible if you can link your family to well known Jews for whom we have a geneology.
For example, I know someone who has a sefer, written by his grandfather, that is a commentary to the Taz titled Shevet Levi. It contains a family tree linking my friend's family to the Taz and linking the Taz to the Baal Shem Tov, and from the Baal Shem Tov they link themselves to Shevet Levi from Temple times.
I have also seen a family tree where -- also using known famous ancestors -- managed to trace their tree to King David using the same method. There, the trick is to find some connection between your family and a Nasi (or Exilarch) -- one of the leaders of the Babylonian Jewish community. Every Exilarch was a descendant of Kings David and Solomon and there were exilarchs until about the 11th century of the common era, according to the Iggeres d'HaRav Sh'irer HaGaon (a fascinating book explaining the development of the Mishna, Tosefta, and Gemara written by the head of the Academy of Pumpedisa).
There are a couple of families listed in the Kohen's Handbook that can in fact trace their lineage back to Aaron. The Rappaport family being the most famous. The Rappaports if I am not mistaken trace their lineage back to Tzadok(As can a couple of other families). There is a Kohanic branch of the Azoulai family that can trace their lineage back to Itamar. I know of three different families that can trace their lineage back to Rabbi Yishmael.
In regards to BA's assertion that lineage is uncertain, especially amongst Sephardim where one is not automatically believed when he says he is a Kohen but that he has to prove lineage, this is only a doubt that there may have been a mamzer or something else such as that that mixed into the line, and thus there is reason to be extra-cautious(sorry don't remember the source in the Sh"A, but it is a basic machloket between the Mechaber and the Rema as to whether we can trust soemone to say they are a Kohen).