Not at all. Upwards class mobility, in the true sense, is essentially impossible within a single generation. Speaking from a British perspective, a child with a poor upbringing is unlikely to be able to convince others, at least on close inspection, that they aren't actually a prole. (How might I know that?)
I am specifically talking about, as I say, "growing up without much money". A poor individual getting from the bottom, or close to it, up to a solid income (say 80th percentile). From there, the wealth is a matter of financial discipline. Some of the 'class' might come, but life is not an act, and Tarquin will always differ.
At least where I'm from (UK), it's a trivial problem in the sense that an equation might be trivial to solve, for an individual. The ingredients are there; an individual must apply themselves to the task at hand.
If you are healthy and of reasonable intelligence, obtain a degree from a good university, and apply those skills to a job, it's quite hard to fail at that catastrophically.
In a collective sense it is impossible, at least the way that society is currently set up. A lot of people have to not be on top.
What confuses people a lot is that they look at these statistical averages and assume that because most of the poor remain poor (and most of the rich remain rich - wealth is sticky), the same likelihoods apply to them.
But if you're actually trying to get out of that situation (most people honestly aren't - they might say they are, but it's not their #1 priority) you're not in that same cohort and you can quite easily be more competitive than those brought up in middle income families.
This is all based on my experience in Britain. In the US it's probably different.
If you're born in a poor _country_ then you probably need to get out first, and I imagine the whole passport visa situation is rather a challenge.
If you're born with severe disabilities, and poor, you could well be shit out of luck. So it goes.
I might be misunderstanding but are you saying that upwards class mobility is easy, specifically from the lowest end?