Can the Wildlings attack twice in a turn?
The Wildlings attack twice.

From the rulebook under "Wildling Attacks"


  Note: In rare circumstances, it’s possible for the wildlings
  to attack twice during one Westeros Phase. This would occur if
  enough Wildling icons were drawn to make the Wildling token
  reach the “12” position on the Wildling track, and the Westeros
  card “Wildlings Attack” was revealed


If the Wildings are defeated on the first attack, the second attack will still happen, but with a threat of 0, in which case they are automatically defeated, but players may still bid to win the boon given to the highest bidder.  From the FAQ:


  Q: If the Wildling Threat token is on the “0”
  space of the Wildling track and a “Wildlings Attack”
  Westeros card is drawn, is it still resolved as normal? Is
  there still the possibility of defeat?
  
  A: The “Wildlings Attack” card is resolved
  regardless of the position of the Wildling Threat
  token. Players still go through the process of
  bidding Power tokens against the Wildlings, but in
  this case, they are guaranteed victory.


Note that the Wildlings do not get an extra attack if the threat level would go to 16 because any advances beyond 12 are ignored:


  Advance Wildlings Track: Count the number of Wildling icons on the three revealed Westeros cards (if any) and move the Wildling Threat token forward that many spaces on the Wildlings track.  If, during this step, the Wildling Threat token reaches position “12”, a Wildling Attack is immediately resolved (ignore any surplus Wildling icons).