Reliable measures of poverty are an essential statistical tool for public policies aimed at reducing poverty. In this paper we consider the reliability of income poverty measures based on survey data which are typically plagued by missing data and measurement error. Neglecting these problems can bias the estimated poverty rates. We show how to derive upper and lower bounds for the population poverty rate using the sample evidence, an upper bound on the probability of misclassifying people into poor and non-poor, and instrumental or monotone instrumental variable assumptions. By using the European Community Household Panel, we compute bounds for the poverty rate in ten European countries and study the sensitivity of poverty comparisons across countries to missing data and measurement error problems. Supplemental materials for this article may be downloaded from the JBES website.