Atomically-detailed molecular dynamics simulations have emerged as one of the most powerful theoretic tools for studying complex, condensed-phase systems. Despite their ability to provide incredible molecular insight, these simulations are insufficient for investigating complex biological processes, e.g., protein folding or molecular aggregation, on relevant length and time scales. The increasing scope and sophistication of atomically-detailed models has motivated the development of "hierarchical" approaches, which parameterize a low resolution, coarse-grained (CG) model based on simulations of an atomically-detailed model. The utility of hierarchical CG models depends on their ability to accurately incorporate the correct physics of the underlying model. One approach for ensuring this "consistency" between the models is to parameterize the CG model to reproduce the structural ensemble generated by the high resolution model. The many-body potential of mean force is the proper CG energy function for reproducing all structural distributions of the atomically-detailed model, at the CG level of resolution. However, this CG potential is a configuration-dependent free energy function that is generally too complicated to represent or simulate. The multiscale coarse-graining (MS-CG) method employs a generalized Yvon-Born-Green (g-YBG) relation to directly determine a variationally optimal approximation to the many-body potential of mean force. The MS-CG/g-YBG method provides a convenient and transparent framework for investigating the equilibrium structure of the system, at the CG level of resolution. In this work, we investigate the fundamental limitations and approximations of the MS-CG/g-YBG method. Throughout the work, we propose several theoretic constructs to directly relate the the MS-CG/g-YBG method to other popular structure-based CG approaches. We investigate the physical interpretation of the MS-CG/g-YBG correlation matrix, the quantity responsible for disentangling the various contributions to the average force on a CG site. We then employ an iterative extension of the MS-CG/g-YBG method that improves the accuracy of a particular set of low order correlation functions relative to the original MS-CG/g-YBG model. We demonstrate that this method provides a powerful framework for identifying the precise source of error in an MS-CG/g-YBG model. We then propose a method for identifying an optimal CG representation, prior to the development of the CG model. We employ these techniques together to demonstrate that in the cases where the MS-CG/g-YBG method fails to determine an accurate model, a fundamental problem likely exists with the chosen CG representation or interaction set. Additionally, we explicitly demonstrate that while the iterative model successfully improves the accuracy of the low order structure, it does so by distorting the higher order structural correlations relative to the underlying model. Finally, we apply these methods to investigate the utility of the MS-CG/g-YBG method for developing models for systems with complex intramolecular structure. Overall, our results demonstrate the power of the g-YBG framework for developing accurate CG models and for investigating the driving forces of equilibrium structures for complex condensed-phase systems. This work also explicitly motivates future development of bottom-up CG methods and highlights some outstanding problems in the field.