I agree with Noah that the two are not mutually exclusive. A quick thought experiment shows this. Imagine a dictator were to set the distribution of incomes so that a few were very rich, many did OK but that a big minority were poor. And he then allocated positions in this distribution on the basis of various psychometric tests for intelligence, laziness and character. Would poverty in this society be due to structures or individuals?
Obviously, to structure - because the dictator has set the structure of incomes so that some are poor.
Obviously also to individuals - because there is a perfect correlation between poverty and low intelligence or other character failings.
The distinction, then, isn't robust.
Another thought experiment might help us distinguish the two. What would happen if everyone were to become intelligent, educated and hard-working?
The "poverty is individual" brigade would say that some form of Say's law would lead to the creation of sufficient good, well-paid jobs to eliminate poverty. The "poverty is structural" band would say that this won't happen, because inequality, deskilling and unemployment are inherent structural features of capitalism, which would condemn some to (relative) poverty even if we were all smart and employable.
What's the empirical evidence here? Immediately, we run into a problem. The research on the links between human capital and aggregate growth, whilst large, is not conclusive (pdf). On the one hand, Erik Hanushek among others thinks there might be big macro-level payoffs to higher cognitive skills. But on the other, there's lots of scepticism about the link between education and growth. Bryan Caplan's reading of the evidence is that the response is "low" - and Middendorf and Krueger and Lindhal agree (pdf). This is consistent to the private payoffs to education being due to signaling - which implies that it is one's relative, not absolute, level of education which determines one's income. If so, better education for all won't raise incomes much.
All this said, two bits of evidence - on top of my priors! - makes me side with Matt in believing that mass upskilling won't eliminate relative poverty.
2. There's no correlation between inequality in human capital and in incomes, either across countries or over time. This is inconsistent with human capital-based explanations for inequality, but consistent with the view that inequality is due to inequalities of power.
Personally, I suspect that the "poverty is individual" view owes a lot to ideology: system justification and the just world fallacy cause people to blame the victim. Actually proving this beyond doubt is, though, perhaps impossible.