A lot of people have been chanting that "Poverty is a choice" mantra lately. It's become a staple of the get-rich-quick-scheme industry (which ironically often helps people to waste even more time and money). I think it's also a rationalization that helps otherwise morally decent people accept the ridiculous elitist philosophy of Ayn Rand.
That said, there are some people who indeed choose to be poor. Some of them even rationally.
But in order to actually choose something, you must have other alternatives, understanding of those alternatives, resources to reach for those alternatives, luck to succeed at those alternatives, and a full understanding of what the actual value is of all alternatives.
You'll find that many of these things are dependent on one's experiences. If you are born amongst people who don't know alternatives or how to get there, you are around people who don't understand the alternatives, and you are never exposed to them in a realistic fashion... and by the time you arem, you have no hope for them... well, you will not be able to choose them.
At the very least we need a strong enough education system to make up for the environmental issues that will affect choices. And even then, I hate to tell you, even if everyone chooses to follow a road out of poverty... some people will still be poor.
The fact is that extreme poverty is, in most cases, temporary. Just as we'd like it to be. But there are conditions that will limit choices and everyone will make mistakes.
The thing that's rough for us middle classers to get, and must be unfathomable to the rich, is that when you hav very few resources you cannot afford to fail. The only way to success is to take risks. Someone who cannot afford to take risks cannot realistically escape poverty.
Sure, take a few thousand people making risks they can't afford... and a few will achieve them and end up becoming more legendary rags-to-riches stories. The rest will not be given credit for trying.

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"Man lives in the sunlit world of that which he believes to be reality. But unseen by most is an underworld, a place that is just as real... but not as brightly lit... A DARK SIDE!"
-opening from Tales From the Darkside