From experiments it is known that magnetostatic interactions between grains strongly affect the magnetic behavior of samples. However, because of the difficulty in predicting the nonlinear behavior, the effect of interactions has been largely ignored from theoretical models. Instead models are often based on noninteracting assemblages. This approximation is valid for certain natural systems, but there are many cases where interactions are known to be important, for example, bacterial magnetosomes found in sedimentary rocks. Using a three-dimensional micromagnetic model, we have conducted a detailed study of the role of magnetostatic interactions on the magnetic properties of assemblages of ideal single domain (SD) grains and cubic grains between 30--250 nm in size. We quantify the contribution of interactions to hysteresis parameters and the Day plot. We show that interactions can strongly affect the magnetic characteristics of a grain assemblage. For example, assemblages of interacting SD grains can plot in the traditional multidomain (MD) area of the Day plot. For grains >100 nm in size, interactions can have the opposite effect, and can cause the hysteresis parameters to shift toward the SD region of the Day plot. In addition to varying grain size, we have also considered various anisotropies, e.g., uniaxial and cubic, and the importance of the alignment configuration of the particle assemblages, i.e., randomly distributed or aligned. It is shown that for assemblages of aligned magnetite particles, that as the interaction spacing is decreased, the SD/MD transition size increases, which may explain why some magnetotatic bacteria possess aligned grains of magnetite above the traditional transition size value of 70 nm. By aligning the anisotropies, the grains become stable SD, and having larger crystals will increase the magnetic signal.