“Bran thought about it. ‘Can a man still be brave if he’s afraid?’
‘That is the only time a man can be brave,’ his father told him.”
After the execution in chapter 1, Ned explains to his son that one cannot be brave unless he is tested, and throughout the book we see that Ned's lesson to Bran applies to more than just bravery. What Ned means is that bravery consists in overcoming fear, not in the absence of fear. What Ned implies is that it is the triumph over fear that is worth praising, and that being without fear isn't particularly praiseworthy in itself. Catelyn raises a similar point when she tells Robb that the Greatjon isn't an ideal choice to lead because he is fearless. She believes it makes the Greatjon act without considering all the risks involved, and in this way being fearless is actually a detriment.
The same premise later applies to the notions of honor and duty that we see primarily in Ned's and Jon's struggles. As Commander Mormont suggests to Jon, one's oath to the Night's Watch is easy to keep if it is not in competition with anything. But it is precisely when that oath is in conflict with, for instance, love for one's family that the oath is proved. In other words, for Jon it is overcoming the temptation to join Robb that proves the strength of his oath. In Ned's case, he finds that it becomes difficult to maintain his honor when it is in conflict with his survival, but that is also when his honor means the most. In King's Landing, he quickly learns that being honorable puts him at a disadvantage, and it is his honorable decision to give Cersei advance warning of his intent to tell Robert about Joff's real father that ultimately leads to his death.