The divergence theorem (or Green-Gauss) gradient scheme is among the most popular methods for discretising the gradient operator in second-order accurate finite volume methods, with a long history of successful application on structured grids. This together with the ease of application of the scheme on unstructured grids has led to its widespread use in unstructured finite volume methods (FVMs). However, the present study shows both theoretically and through numerical tests that the common variant of this scheme is zeroth-order accurate (it does not converge to the exact gradient) on grids of arbitrary skewness, such as typically produced by unstructured grid generation algorithms. Moreover, we use the scheme in the FVM solution of a diffusion (Poisson) equation problem, with both an in-house code and the popular open-source solver OpenFOAM, and observe that the zeroth-order accuracy of the gradient operator is inherited by the FVM solver as a whole. However, a simple iterative procedure that exploits the outer iterations of the FVM solver is shown to effect first-order accuracy to the gradient and second-order accuracy to the FVM at almost no extra cost compared to the original scheme. Second-order accurate results are also obtained if a least-squares gradient operator is used instead.