I think it's a bit more nuanced than that. As I understand it, happiness increases for most people as their income increases. However, this doesn't mean that a person is happy overall since there are other factors. So, it's not that money can buy happiness in a binary sense, but it's a factor and often a significant one.

The article even ends with this quote from one of the authors of the study (emphasis added):

“Money is not the secret to happiness, but it can probably help a bit.”