Hypertree decompositions, as well as the more powerful generalized hypertree decompositions (GHDs), and the yet more general fractional hypertree decompositions (FHD) are hypergraph decomposition methods successfully used for answering conjunctive queries and for solving constraint satisfaction problems. Every hypergraph H has a width relative to each of these decomposition methods: its hypertree width hw(H), its generalized hypertree width ghw(H), and its fractional hypertree width fhw(H), respectively. It is known that hw(H) k can be checked in polynomial time for fixed k, while checking ghw(H) k is NP-complete for k 3. The complexity of checking fhw(H) k for a fixed k has been open for over a decade. We settle this open problem by showing that checking fhw(H) k is NP-complete, even for k = 2. The same construction allows us to prove also the NP-completeness of checking ghw(H) k for k = 2. After proving these hardness results, we investigate meaningful restrictions, for which checking for bounded ghw and fhw is easy. In particular, we study classes of hypergraphs that enjoy the bounded edge-intersection property (BIP), the more general bounded multi-edge intersection property (BMIP), the bounded degree property (BDP) and the bounded VC-dimension. Given the increasing interest in using such decomposition methods in practice, a publicly accessible repository of decomposition software, as well as a large set of benchmarks, and a web-accessible workbench for inserting, analysing, and retrieving hypergraphs are called for. We address this need by providing (i) concrete implementations of hypergraph decompositions (including new practical algorithms), (ii) a new, comprehensive benchmark of hypergraphs stemming from disparate CQ and CSP collections, and (iii) HyperBench, our new web-interface for accessing the benchmark and the results of our analyses. In addition, we describe a number of actual experiments we carried out with this new infrastructure.