Many cellular functions involve protein complexes that are formed by multiple interacting proteins. Tandem Affinity Purification (TAP) is a popular experimental method for detecting such multi-protein interactions. However, current computational methods that predict protein complexes from TAP data require converting the co-complex relationships in TAP data into binary interactions. The resulting pairwise protein-protein interaction (PPI) network is then mined for densely connected regions that are identified as putative protein complexes. Converting the TAP data into PPI data not only introduces errors but also loses useful information about the underlying multi-protein relationships that can be exploited to detect the internal organization (i.e., core-attachment structures) of protein complexes. In this article, we propose a method called CACHET that detects protein complexes with Core-AttaCHment structures directly from bipartitETAP data. CACHET models the TAP data as a bipartite graph in which the two vertex sets are the baits and the preys, respectively. The edges between the two vertex sets represent bait-prey relationships. CACHET first focuses on detecting high-quality protein-complex cores from the bipartite graph. To minimize the effects of false positive interactions, the bait-prey relationships are indexed with reliability scores. Only non-redundant, reliable bicliques computed from the TAP bipartite graph are regarded as protein-complex cores. CACHET constructs protein complexes by including attachment proteins into the cores. We applied CACHET on large-scale TAP datasets and found that CACHET outperformed existing methods in terms of prediction accuracy (i.e., F-measure and functional homogeneity of predicted complexes). In addition, the protein complexes predicted by CACHET are equipped with core-attachment structures that provide useful biological insights into the inherent functional organization of protein complexes. Our supplementary material can be found at http://www1.i2r.a-star.edu.sg/~xlli/CACHET/CACHET.htm ; binary executables can also be found there. Supplementary Material is also available at www.liebertonline.com/cmb.