a song of ice and fire - Can a Bastard, under any circumstance, become a Lord or King in Game of Thrones

As stated in the title of the question, can a Bastard, under any circumstance whatsoever(except an armed rebellion), become a Lord or for that matter, a King in Game of Thrones.

I am in the middle of Season 1 & Book 1 so ANY SPOILERS ARE PUNISHABLE BY DEATH

I am looking for a generic answer but your answer can include the example of Eddard Stark & Jon Snow but only hypothetically. NO SPOILERS
I hear there is a spoiler on Jon Snow floating around on the internet & on the Stark family. I havn't read them neither intend to. So please answer this question generically