Yes the player can draw cards even if he has 5 (or more) cards already in their hand.
Page 5 of the rules:
or b) Draw a card from his deck.
If a player already has 5 or more cards in his hand, however, he may not choose to draw a card (although he may still play cards which grant additional cards (emphasis mine)). Apart from this restriction, there is no hand size limit.
That rule says you cannot draw a new card from the deck at the start of the round, but can play cards that give you extra cards beyond the 5 in your hand. I would assume that the same thing is true with a Breaking News card as well.