While for many methods in computational mechanics there is often a way of estimating the error in the computed solution, the situation for the Material Point Method (MPM) and also Particle in Cell Methods is very different in that there has been comparatively little work on computed error estimates for these methods. This work is concerned with building upon an approach introduced by the author for the Material Point Method.The approach, makes precise the sources and forms of the different MPM errors. There is then a need to estimate these errors through computable estimates of the different errors in the material point method. The approach used involves linearity-preserving extensions of existing methods, which allows estimates of the different spatial errors in the Material Point Method to be derived based upon nodal derivatives of the different physical variables in MPM. . The use of these estimates of the spatial error makes it possible to measure the growth of errors over time. A number of computational experiments are used to illustrate the performance of the computed error estimates for a variety of MPM methods.