I think there still is the ability to change class in the USA, but not everyone is going to take advantage of it. People may think they want to "move up", but it involves making tough choices. Perhaps to some extent people feel more comfortable staying in the class they were born in.
I base this on my family. Father--maintenence man; Mother--Stay at home mom; working class all the way. My siblings all chose to follow this pattern. My sisters all wanted to stay at home with their kids, but each ended up divorced (multiple times) and working out of the home. Brother quit high school and learned a trade. This was back in the '70s, which I believe was the last decade when a working class kid could reasonably expect to find a living wage job with a high school diploma (or none, even.)
I got the college degree, postponed marriage, postponed children. Married, no divorce. My husband and I both found white-collar jobs in government, not much in terms of wages, but good pensions and health care. I would have prefered to stay home with my kids, but I compromised with a half time job.
None of my sibs are stupid or lazy, but the difference between "right choices" and "wrong choices" decides where you go on the ladder. From my observations, these are the "wrong choices":
education: High school diploma, GED or dropout
Marriage and children at a young age
Multiple marriages.
For women: marrying one loser after another, because you feel sorry for them. Or marrying some idiot because he is the father of your children. Marrying anyone and thinking you can improve them. Relying on men to support you, in short, and not preparing for a career in case you end up having to work out of the home.
Ensuring future ill-health by smoking, lack of exercise, alcoholism and drug abuse.
A history of petty crime, even if you've "gone straight"
Mental illness, depression, low self esteem
A basic attitude that higher attainment is impossible, and people who do attain are "cheating" somehow, through getting unfair advantages.
A fatalistic attitude that you might as well spend what you've got now and get a little pleasure from it, because you'll never save enough to make a difference.
Obviously there are exceptions to these points, but overall I'd say these are the things that sank my sibs' expectations. Unfortunately, I'd also have to say that their role models were my mom and dad.