Many dynamic problems have constraints on accumulation. For example accumulation capacity is finite for water, oil, natural gas, and electricity. We present a dynamic accumulation model with limited capacity, and provide foundations for its numerical solution. We prove that time iteration (TI) operators defined on either the value function or its Euler equation are convergent. The influential work of Judd (1998) emphasizes the need for a better understanding of the stability properties of fixed-point iteration (FPI) operators, and shows that FPI operators are not necessarily locally convergent, even for linear models. Deaton and Laroque (1995) and Cafiero et al. (2011) report stability problems in their econometric estimates using FPI operators. We prove the FPI operator does not necessarily converge in models of this type. However we prove convergence in a subset of models with linear marginal utility. Our results can be useful for solving models designed for policy evaluation based on either calibrations or structural econometric estimations that require numerical solutions.