In developed countries, income-based measures of poverty identify large shares of the population (17.1% of the EU at the 60% threshold in 2018). Defined as such, the 'poor' are very diverse in terms of the severity of their situations. In this work, we propose an indicator that targets those of the income-poor that experience harsher living conditions. We show that it not possible to achieve a significant targeting of deep poverty using income data only: at the lowest levels of income, it ceases to be correlated with other indicators of economic distress. The insufficiency of income to identify deep poverty still holds when imputed rents of owners or remunerations in kind are taken into account. Adding material deprivations, however, allows to take into account non-income components of a household's standard of living, such as wealth, health or occupational status. Using European data on households income and living conditions, we show that households who combine low income and a high number of material and social deprivations suffer from a more severe situation of poverty, in terms of subjective well-being and other measures of economic distress, and stay longer in poverty.