Apparently, both the Mersenne Twister and XorShift (which seem to be two of the popular RNGs people use these days) can't pass all statistical tests. (The Mersenne Twister is pretty huge as well.)
Looking at the PCG paper, it shows that a basic XorShift generator doesn't pass statistical tests as it gets bigger. Basically, there are some tests it never passes no matter how big it gets.
But a tiny change (XorShift*) is enough to get a generator that passes empirical statistical tests.
The page about other RNGs has lots of good info about this stuff.