Generating ``good'' random numbers can be a major problem in Monte Carlo simulations. Many basic random number generators (RNGs) supplied with computers (especially parallel computers) are not sufficient. Particularly true in the past, but unfortunately still true far too often today!
There are many statistical tests for uniform distribution, correlations, etc.
Monte Carlo measurement of the energy of the 2-d Ising model is a good
test --- the result is known exactly for 
lattice.
Some RNGs pass statistical tests, but fail the Ising model test!
Even some ``good'' RNGs having adequate randomness properties are not good enough for Monte Carlo simulation because their period of repetition is too small.