Let X1, X2, …, Xn be a sequence of independent or locally dependent random variables taking values in ℤ+. In this paper, we derive sharp bounds, via a new probabilistic method, for the total variation distance between the distribution of the sum ∑i=1nXi and an appropriate Poisson or compound Poisson distribution. These bounds include a factor which depends on the smoothness of the approximating Poisson or compound Poisson distribution. This “smoothness factor” is of order O(σ−2), according to a heuristic argument, where σ2 denotes the variance of the approximating distribution. In this way, we offer sharp error estimates for a large range of values of the parameters. Finally, specific examples concerning appearances of rare runs in sequences of Bernoulli trials are presented by way of illustration.