Daenerys is the product of years and years of inbreeding. Her family married brothers to sisters to keep them pure.
Arya & Jon and Tyrion & Jaime.
Book theory below.
I think that Jon is not a Bastard. I think Jon is really Ned's nephew. I think that Ned's sister willingly went with Rhaegar Targaryen and married him, despite him already being married. I think that when Ned and the others got to the tower in the mountains where his sister was being guarded by Kingsguard it was because his sister had had a baby. I can think of no other reason that the Kingsguard would be there. Their mission is to protect the King and his family. Ned's sister was not royal. I think that Ned claimed the baby as his own to shield Robert from the pain and to keep him from killing the "Targaryen" baby. We may never know since Ned died. Only one other person returned from the tower. That is the Reed children's father, Howland Reed, The Lord of the Crannogmen.