Yes. And no. The research suggests that it really depends on what you mean by "happiness."
In 2010 a major study was published by economist Angus Deaton and psychologist Daniel Kahneman that looked at the relationship between money and happiness by analyzing the Gallup surveys of 450,000 Americans.
The study considered two types of happiness:
1. Day-to-day contentment (emotional well-being)
2. One's general sense of satisfaction with their place in the world ("life assessment")
They found that while income correlated with more positive "life assessments," it only predicted increases in day-to-day contentment up to a point. That point was $75,000 dollars. Beyond that, income did not correlate with or predict levels of emotional well-being.
If you Google this study you'll find lots of intelligent critiques and questions about the $75,000 cut-off. "The Gallup surveys were from 2009, so the cut-off must be higher now," "it just so happens that the national average income for 2009 was around $75,000. However, the cost of living in places like the Bay area and Manhattan are significantly higher than in parts of Kansas or Texas. Does that mean the cut-off is higher in these places?"
In my opinion, these critiques (though totally legitimate and important) shouldn't overshadow the message, which is that after a certain point, making more money is not going to have a significant effect on your day-to-day happiness. It will continue to positively affect how we feel generally about our place in the world, but this will not likely be reflected in our daily mood.
So what other factors correlate with emotional well-being? Interestingly, they're often things that are at odds with the sacrifices necessary to make more money.
Consider commute times. Research at Canada's University of Waterloo showed a direct correlation between commute time and well-being. The results, published in the World Leisure Journal, suggest that those with the longest commutes report the lowest levels of life satisfaction. These findings support those published in the American Journal of Preventative Medicine, in which longer commutes correlated with poor cardiovascular and metabolic health.
In addition to commute times, we know things like exercise, meditation, and diet affect our mood. Again, many of these priorities are often compromised so that we can work harder and earn more.
The truth is, higher income does predict levels of day-to-day happiness, up to a point. What that exact point is might be harder to determine than the research suggests, but it appears to be around whatever the average income is for your geographic location. Once you've reached that point, however, earning more isn't likely to affect your emotional wellbeing. It might even interfere with your ability to attend to other factors that have been shown to directly impact your quality of life, such as getting good sleep, spending time with friends, and exercising.
So if you're trying to decide whether or not to take that promotion or new job, consider what it will cost you. Will it affect your commute, your time at home, your ability to attend to your physical health? If you are already financially comfortable, the extra money and prestige might not actually make you any happier.