We apply several methods to estimate quantum mechanical partition functions for one-dimensional oscillators using only limited information, namely, the classical dissociation energy and the expansion of the potential through the quartic term in displacements from the minimum. The methods considered are based on the Dunham expansion or the Pitzer-Gwinn method. Various corrections are considered to try to ensure accuracy both at low temperature, where the partition function is most sensitive to the zero-point energy, and at high temperature, where it is essential to have approximate energy levels that increase in a reasonable fashion all the way to the dissociation energy. A method based on a Padé approximant modification of the Dunham expansion is found to provide a useful compromise of accuracy and simplicity for a wide variety of cases. We also show that the Pitzer-Gwinn method may be used successfully even when only limited information is available.