We eventually figure out:
It was Joffrey who sent the catspaw to assassinate Bran using the Valyrian steel dagger. It wasn't Jaime, Cersei or Tyrion that wanted Bran dead.
My question is: why wouldn't Cersei and Jaime try to kill Bran after realizing that he'd awaken after the fall? Weren't they afraid that he'd remember what happen and expose them?
They would have liked to kill him, as discussed in Chapter 9 of AGOT, but when would they have gotten the chance? At that point Bran was being watched 24/7 by his mother, and both Jaime and Cersei are prominent royal guests. It may not have been possible for them to make suitable plans without exposing themselves. At that point the safer play may have been to just hit the road for King's Landing as soon as possible.
Also, by that point the Starks were suspicious about what happened since Bran had never fallen before, and Catelyn received a letter from Lysa blaming the Lannisters for Jon Arryn's death. Granted, Jaime and Cersei did not know about the extent of this, but it may have added to the tension at Winterfell. All the more reason for them to leave before the situation gets any worse.
I forget where it is said, but Cersei sees Bran and realizes that he does not remember anything.
Also, these theories:
They also have more important things to do. The rumor is already out so them trying to silence him would be illogical.
More to the point they have bigger fish to fry, an he doesn't intimidate them.
Jaime Lannister: I think we can outfox a ten year old.
Cersei Lannister: And my husband?
Jaime Lannister: I'll go to war with him if I have to... they can write a ballad about us, "The War for Cersei's Cunt!" [Cersei slaps Jaime. He chuckles, and she tries to slap him again, but he grabs her arm and holds her from behind, firmly
Cersei Lannister: Let me go...
Jaime Lannister: Never.
Cersei Lannister: Let me go...
Jaime Lannister: [whispers reassuringly in Cersei's ear] The boy won't talk. And if he does, I'll kill him, Ned Stark, the king, the whole bloody lot of them until you and I are the only people left in this world. [Cersei and Jaime quietly sink into a loving embrace]
The two are caught having sex by Bran Stark, and when Cersei insists that they must do something to prevent the boy from telling anyone, Jaime immediately pushes him off a tower.[8] She later berates Jaime for his impulsiveness, arguing that attempting to kill the boy was foolish when they could have simply intimidated him into silence.
Its futile killing him when everyone already knows what he will say. Plus, they might think he is scared. Most of his family is dead or missing.
It wasn't a Lannister? Isn't Joffrey 100% Lannister? – Shevliaskovic Aug 26 '14 at 6:24