In theory, it is possible that the persistent poverty that has emerged in many transition economies, is attributable to underlying, non-convexities in the dynamics of household incomes - such that a vulnerable household will never recover from a sufficiently large, but short-lived shock to its income. This happens when there are multiple equilibria in household incomes, such that two households with the same characteristics, can have different incomes in the long run. To test the theory, the authors estimate a dynamic, panel data model of household incomes, with non-linear dynamics, and endogenous attrition. Their estimates, using data for Hungary in the 1990s, exhibit non-linearity in the income dynamics. The authors find no evidence of multiple equilibria. In general, households bounce back from transient shocks, although the process is not rapid.