We study convergence properties of nonlinear systems in the presence of 'almost Lyapunov' functions which decrease along solutions in a given region not everywhere but rather on the complement of a set of small volume. The structure is quite general except that the system dynamics never vanishes in a region that is away from the equilibrium. It is shown that solutions starting inside the region will approach a small set around the origin as long as the volume where the Lyapunov function does not decrease fast enough is sufficiently small. The main theorem of this paper is established by tracking the change of Lyapunov function value when the solution passes through the above mentioned volume and finding an upper bound of the volume swept out by a neighborhood along the solution before it can achieve an overall gain in its Lyapunov function value. The result shows that the convergence rate is traded off against the size of such small volume that the system can have. In the end a non-trivial example where our theorem is applicable is demonstrated.