Honestly, it depends on how you define "happiness".
My family struggled with money for most of my life, not because my parents are irresponsible people, but because the economy sucks and businesses lay people off and once you're in debt, it's really ******** hard to get out. My brother and I spent most of our childhood moving from place to place. We had two different houses foreclosed on us. That kind of thing is borderline traumatizing when it happens repeatedly. You never know what's really yours, or how long you're going to have it.
For me, money is security. It's a cushion. Having money makes me feel safe, because I know I can stay where I am. I don't have to worry about moving, or having the electricity shut off, or not being able to afford things when I need them. And that peace of mind, that ability to sleep at night free of anxiety over money, is happiness to me.
So in a way, at least for me, yes. Money can buy happiness.