We use a lattice model with orientation-dependent interactions to study network-forming fluids, such as water and silica. Bonds can form between pairs of molecules that have correct mutual orientation and correct separation; they can be weakened by the presence of other molecules sufficiently close to a bonded pair. These interactions give rise to competition between bonded states of low energy, density, and entropy and nonbonded states of high energy, density, and entropy. By suitable choice of parameters, the mean-field solution of the model yields the two different scenarios (stability limit conjecture and second critical point) that have been proposed to explain the anomalous behavior of supercooled water. The model’s generality suggests that similar behavior can occur in other network-forming fluids.