Statistical tests are vital, but passing such tests is no guarantee that a generator will perform well for a certain application. Tests that are specific to certain applications can be more effective.
In the case of Monte Carlo simulations, there are a number of problems (e.g. 2-D Ising model, simple percolation and random walk models) for which exact results are known. It is thus possible to compare Monte Carlo results with exact results.
These ``Monte Carlo tests'' provide some of the best empirical tests of randomness.