Are there flaws in (cryptographic) science/engineering/software/research due to pseudo random generators?
There are a few examples:


RANDU is probably the quintessential example of a bad pseudorandom generator. It consisted of the sequence $x_{i+1} = 65539\cdot x_i \bmod{2^{31}}$, which is always odd and whose sequential triples are highly correlated. It was implemented by IBM in its software, and so it was used all over the place. It has been suggested that research results relying on RANDU are not reliable, but I could not track down any concrete example of this.
Another interesting example is that of 1992 Ferrenberg et al., in which a pseudorandom number generator that passed the statistical tests common of the time, but nevertheless resulted in incorrect results in a real Monte Carlo Ising-Model simulation. The generator in question was R250, a very simple generalized feedback shift register.


There are most likely many other examples incorrect results that simply haven't been checked or reproduced.