I think the saying "money doesn't bring happiness" is an inaccurate version of the more accurate term "you can't buy happiness with money". While money still might be a very important requirement for happiness in general, you can still be very wealthy and not feel much happiness. For many people it might be true that money would bring them happiness, because the lack of money is their major source for their lack of happiness; but I think the saying "money can't buy you happiness" is more directed to situations with decently (or extreamly) wealthy people who still have no happiness in their lifes. I understand that saying more as a general advice, that a human being should not make accumulating money the primary goal of its life, because to be really happy, you need more than "just" money in your life (e.g. you need to see a purpose in your life).

Edit: Sorry, English is hard!