In 2013 more than 70% of Americans did not pay off the credit card bills every month - most credit card holders carry a balance and try to pay it off over time.
One study was done to disprove Dave Ramsey's snowball method. However, it actually did the opposite. It showed that even though paying off the higher interest rates first saves money over time and the people in the study knew it, people in the study were actually more driven by their emotions and they stuck with the plan that allowed them to feel success. If people don't follow the advice (like pay down debts based on interest rates), it does them no good.
Here is another study done of 6,000 debt holders which found that those who paid off smallest balance first were more likely to pay off all their debts.
And don't forget, Dave doesn't say to ignore the higher interest rates forever. He basically says, small things first - you'll get to the higher interest rates soon.