OK as we all know The Septim dynasty is quite large But heres a theory.
Now Imagine an emporor has 5 children -Atleast one of those children will get married and have more children, thus extending the family tree, now if this has happend in the hundreds of years beetween Tiber Septim and Martin - should there not be hundreds of septims (Dragonbornes) Still alive should martin not have lots of 1st second and third cousins etc.
Could our skyrim dragonbourne be one of theese?
Why did uriel only have his sons why not his many uncles cousins etc.
Major plot error:P
Also could this theory explain skyrims dragonbournes existence!

I don't really see a plot error here...
Uriel was an old man, thus his uncles would most likely be dead. Who's to say he even had any cousins/siblings? Just because he had more than one son, doesn't mean everyone in the line had to have that.

It has already been confirmed that you are the last of the dragonborns, and considering both Martin and his brothers were killed without leaving any heirs, it's highly unlikely the line continued any further, and thus ended when Martin sacrificed himself. After the Septim line was broken, the Mede line took it's place as the emperors of the empire. Titus Mede is the first of the Mede line to rule, and the only known emperor of that line. (the fate of Attrebus Mede, the son of Titus, is still uncertain.)

But yes, there were other dragonborns, but not only the Septim line can be dragonborns - dragonborns are the dragon hunters, but how many, and who exactly they were, are uncertain.

The Septim line is from Skyrim, so dragonborns have always been present in Skyrim.