We use the reverse Monte Carlo (RMC) method to critically evaluate the structural information content of diffraction data on bulk water by fitting simultaneously or separately to X-ray and neutron data; the O-H and H-H, but not the O-O, pair-correlation functions (PCFs) are well-described by the neutron data alone. Enforcing at the same time different H-bonding constraints, we generate four topologically different structure models of liquid water, including a simple mixture model, that all equally well reproduce the diffraction data. Although earlier work [Leetmaa, M.; et al. J. Chem. Phys. 2008, 129, 084502] has focused on tetrahedrality in the H-bond network in liquid water, we show here that, even for the O-O-O three-body correlation, tetrahedrality is not strictly defined by the data. We analyze how well two popular MD models (TIP4P-pol2 and SPC/E) reproduce the neutron data in q-space and find differences in important aspects from the experiment. From the RMC fits, we obtain pair-correlation functions (PCFs) that are in optimal agreement with the diffraction data but still show a surprisingly strong variability both in position and height of the first intermolecular (H-bonding) O-H peak. We conclude that, although diffraction data impose important constraints on the range of possible water structures, additional data are needed to narrow the range of possible structure models.