This is an interesting question. I've seen this idea bandied about personal finance blogs and forums before, and always thought it was a bit nutty, since, as noted in the comments to this question, paying off the debt with the highest interest rate first will minimize both total payments and time until you are debt free. Despite this, it is a notable claim, and one that is widely repeated (at least in my experience).
It turns out this strategy has been studied very recently by the US "National Bureau of Economic Research" (NBER), which appears to be a think-tank devoted to economic policy issues. They claim to be "non-partisan", though I have no way of verifying this.
The introduction of their working paper (page 1) provides a good synopsis of the claim and reasons why it might be true. The claim was popularized by radio host Dave Ramsey.
This paper also cites a 2012 analysis of data gathered from a debt collections agency, which showed that the snowball method was actually an extremely effective strategy for people in arrears. The number of accounts closed so far turns out to be a strong predictor of closing more accounts, regardless of the dollar amounts on the individual accounts. The dollar amounts paid off so far, in contrast, were not predictive.
The 2012 study already answers the question empirically (yes, at least for people sent to collections agencies, the snowball is effective), but the 2014 study investigates why this might be in much more detail, and proposes a full mathimatical model for the actions of human agents in paying off the debt (it looks like a great read, and doesn't seem to be paywalled). They also offer experimental results on a different set of tasks, showing that humans both i) perform tasks faster (voluntarily) when the tasks are ordered from fastest to slowest instead of slowest to fastest or randomly ii) humans are more likely to complete the entire set of tasks when they are ordered from shortest to longest than the other way round (71% completion rate vs. 48%!), or randomly (71% vs. 58%!). Interestingly, although past research found that, given the choice, humans prefer to pay off debt by ordering it from smallest to largest, in this study humans who were allowed to order their tasks actually preferred to do that largest ones first, even though this was less efficient!