This paper introduces and demonstrates four new statistical attacks against the Kirchhoff-Law-Johnson-Noise (KLJN) secure key exchange scheme. The attacks utilize compromised random number generators (RNGs) at Alices/Bobs site(s). The case of partial correlations between Alices/Bobs and Eves probing noises is explored, that is, Eves knowledge of Alices and Bobs noises is limited but not zero. We explore the bilateral situation where Eve has partial knowledge of Alices and Bobs RNGs. It is shown that in this situation Eve can crack the secure key bit by taking the highest cross-correlation between her probing noises and the measured voltage noise in the wire. She can also crack the secure key bit by taking the highest cross-correlation between her noise voltages and her evaluation of Alices/Bobs noise voltages. We then explore the unilateral situation in which Eve has partial knowledge of only Alices RNG thus only those noises (of Alice and Eve) are correlated. In this situation, Eve can still crack the secure key bit, but for sufficiently low error probability, she needs to use the whole bit exchange period for the attack. The security of the KLJN key exchange scheme, similarly to other protocols, necessitates that the RNG outputs are truly random for Eve.