Conformal proton radiation therapy requires accurate prediction of the Bragg peak position. Protons may be more suitable than conventional x-rays for this task since the relative electron density distribution can be measured directly with proton computed tomography (CT). However, proton CT has its own limitations, which need to be carefully studied before this technique can be introduced into routine clinical practice. In this work, we have used analytical relationships as well as the Monte Carlo simulation tool GEANT4 to study the principal resolution limits of proton CT. The noise level observed in proton CT images of a cylindrical water phantom with embedded tissue-equivalent density inhomogeneities, which were generated based on GEANT4 simulations, compared well with predictions based on Tschalar's theory of energy loss straggling. The relationship between phantom thickness, initial energy, and the relative electron density resolution was systematically investigated to estimate the proton dose needed to obtain a given density resolution. We show that a reasonable density resolution can be achieved with a relatively small dose, which is comparable to or even lower than that of x-ray CT.