The analysis of nested structures in sets of species assemblages across different sites or in networks of interspecific interactions has become common practice in ecological studies. Although new analyses and metrics have been proposed, few studies have scrutinized the concepts that subtend nestedness analysis. We note two important conceptual problems that can lead to terminological inconsistencies and flawed interpretations. First, the thermodynamic analogy that underlies the most common metric of nestedness, matrix temperature, is flawed and has led some authors to incorrect interpretations. Second, the term "anti-nestedness" is a potential source of confusion and inconsistencies. We review four concepts for anti-nestedness and examine how distinct they are. "Anti-nested" matrices, i.e. less nested than expected by chance, may result from different ecological processes and show distinct structural patterns. Thus, there is no single unequivocal opposite of nestedness to be represented as "anti-nestedness". A more profitable approach is to designate and test each distinct non-nested pattern according to its specific assumptions and mechanistic hypotheses.1164716722