To what extent can the structure of a disordered heterogeneous material be reconstructed using limited but essentially exact structural information about the original system? We formulate a methodology based on simulated annealing to reconstruct both equilibrium and non-equilibrium dispersions of particles based only on correlation functions which statistically characterize the system. To test this method, we reconstruct dispersions from the radial distribution function (RDF) associated with the original system. Other statistical correlation functions are evaluated to compare how well the reconstructed system matches the original system. We show that for low-density systems or high-density systems with little particle aggregation, our reconstruction from the RDF reproduces the system fairly well. However, for dense systems with extensive clustering, RDF information is somewhat inadequate in being able to reconstruct the original system. We also show that one can produce a system with an RDF that is similar to the reference system, but with appreciably different structure. Finally, system-size effects are analyzed analytically.