Let X₁, X₂, …, X_n 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=1ⁿX_i 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.