Just a thought that I had:

I was doing the Ghost of Old Hroldan quest, where you return a sword for a ghost who keeps calling you Hjalti, one of Tiber Septim's old nicknames, and it suddenly occured to me that the ghost might be right. Why would the ghost mistake you for Tiber Septim after you sleep in Tiber Septim's bed? Maybe because you are Tiber Septim.

Consider: in Morrowind, you turn out to be the reincarnation of Nerevar. Unaware of it at first, you start exhibiting skills far beyond those of any ordinary mortal, and eventually become Hortator of every House. Eventually even Dagoth Ur accepts you as the second coming of Nerevar, a man he knew very well as a close personal friend. Here we find the Dragonborn appearing with a unique skill which everyone keeps saying has not been seen since Tiber Septim. He appears at a time when the worship of Talos has been outlawed and everyone is saying that Talos is either destroying Skyrim or about to save it as a result. As in Morrowind, you immediately start gathering honours and titles like ordinary mortals collect postage stamps, and eventually stick a sword into the greatest dragon which ever existed. Tsun even hints that you're no ordinary mortal, commenting on how the living simply aren't allowed into Shor's Hall... right before he lets you into Shor's Hall.

Well, you are dragonborn after all, a dragonborn that has absorbed the power from numerous dragons, you being exceptionally skilled and powerful is no surprise. You being a reincarnation of Tiber Septim seems possible, but isn't exactly confirmed by these few vague references. Certainly an interesting theory, however you have to keep in mind that the ghost mistaking you for Tiber Septim could simply be because he senses that you both have been blessed by Akatosh, not necessarily because you are his reincarnation.

Yes.

Whenever you find yourself asking a TES lore question like "Could it be that..." or "Is it possible that...", the answer is typically going to be yes.

After reading that, I tried to think of something where the answer would always be no... I honestly can't think of anything. In a universe where magic is a thing, where time can become non-linear and split up into multiple branches and then come together later, or where even the entire existance of mortals could all just be a dream, it's as good as impossible to find a theory that isn't possible.

True.

The Greybeards name you both the Stormcrown (Talos) and the Ysmir, I feel like it can be safely assumed that the Dovahkiin is a Talosine.

The Last Dragonborn is as much Talos as Pelinal is Shezzar. Distinct entities, but mythic echoes.

the LDB is not Tiber Septim, the LDB is a Shezzarine

Pelinal vehemently denied being Shezzarine, though.

The Dragonborn...is pretty much the Dragonborn. Same with the Nerevarine - the Moon-and-Star doesn't kill you, and the enchantment placed upon it means you've got Nerevar's soul.

I don't believe the Dragonborn is Talos walking again, however. The Shrines to Talos still function appropriately, so Talos the divine entity has not made himself manifest. That's not to say he's not Tiber Septim, the mortal component of the god Talos, made manifest however.

Though in all likelihood you're the last of Antiochus Septim's long and sordid line with no clue to your heritage until Alduin's return puts the fire (metaphorically speaking) back into your veins.

Dude. Pelinal is the arch-Shezzarine. Moths and vehemence nothwithstanding.

And that Antinochus thing is quite unlikely. Dragonblooded is not the same as Dragonborn. Being a Septim heir doesn't make you able to Shout, and being able to Shout certainly doesn't necessitate that you are a Septim heir. Akatosh has agency of his own here.

I'm pretty sure Dragonblooded and Dragonborn are the same thing. http://elderscrolls.wikia.com/wiki/The_Book_of_the_Dragonborn specifically says that being a Dragonborn is not hereditary and it uses the term "Dragonborn" and "Dragonblooded" synonymously.

While I don't disagree with the over all statement (the Septims are just flat out weird), the idea that Dragonblooded /=/ Dragonborn has been soundly quashed by multiple characters and quests in Skyrim. Esbern (and IIRC, Paarthurnax) refer to you as Dragonblooded interchangeably with Dragonborn enough times to at least prove that there's no discernible difference between the two.

Indoodidly, Paarth declares you've got authentic dragonblood coursing strong and reptilian in your veins after he tastes your delicious thu'um secretions.

Heck, the Amulet of Kings is enchanted in such a way that only those of the Dragon's blood can wear it. Meaning everyone who ever has worn the Amulet (Which, coincidentally, does not mean all the Septim Emperors have worn it - "Most people think it's the Red Dragon Crown, but that's just jewelry...") is a Dragonborn. Meaning that, yes, Uriel Septim VII inherently knew how to Shout.

Unfortunately, since the end of the Dragon War, Dragons by which to consume souls and thus unlock the inherent knowledge of the Thu'um haven't exactly been available (Though I do like that M'aiq's response to where Dragons are in Morrowind turns out to be absolutely true.). And I don't see many of the royal blood forswearing everything to go study with the Greybeards for years on end just to learn one part of a single shout.

Main reason I mentioned Antinochus is his history is the most likely source of "divergence" in blood lines that not only gets the Dragon's Blood into the Fourth Era, but also survives the purge the Mythic Dawn carried out at the end of the Third. And the only reason I mention the Septims at all is because they are the most...dynamic...of Dragonborn within the franchise, and the most prominent. If the Last Dragonborn is indeed the last, then it seems likely that his ancestry had at least passing contact with the dynasty.

Or it might not have. Mankar Cameron also wore the amulet, which showcased others with Dragon's Blood did exist by the end of the Third Era.

reading through Mankar Camerons commentaries, it appears he may have found a way to artificially become Dragonborn without being blessed at birth, and that may be why he was able to wear the Amulet of Kings:

excerpt from http://www.uesp.net/wiki/Lore:Commentaries_on_the_Mysterium_Xarxes:

The LDB is not Tiber Septim. He is the last one in a long line predating the third empire...

That's a great find.

Excluding that Akatosh/All-maker/ convenient powers that be didn't up and decide "Yup, (s)he'll do" and bestow it on some poor schmuck at some point and time. Having a line survive for that long (when there's more evidence then not that blows the hereditary issue out of the water) unbeknownst to the world is a feat in of itself, and is a tall task to make workd provided you're not roleplaying it that way to begin with.

As for being a avatar of Tiber, well...eh. Stranger things have happened. Not something I'd personally role with, but its not outside the realm of possibility.