I find this question a bit tricky. I've heard the saying that "the rich also cry" which makes me want to conclude that money can't happiness. I've read in the news where billionaires commit suicide then I think, with all that money, why didn't they buy the happiness that they deserved.
My conclusion was that money can buy a lot but definitely not everything you want. There are priceless things like love, respect, integrity, trust etc that no matter how much you've got, it doesn't guarantee getting them.
Look at it this way again. What if the one thing that can make someone truly happy is that expensive treatment abroad, that house to finally call home, an education, food on the table. The things that one cheque can take care of. If you offered someone that fulfillment, can't we say that money has bought happiness?