Mixing of ionic liquids provides new opportunities for their tuning, enabling the applications of ionic liquid mixtures to expand. At the same time, the genesis of the fundamental properties of ionic liquid mixtures is still poorly understood. In this study we carried out a molecular dynamics simulation of binary mixtures of 1-buthyl-3-methylimidazolium hexafluorophosphate, 1-buthyl-3-methylimidazolium bis(trifluoromethylsulfonyl)imide, and 1-buthyl-3-methylimidazolium tris(perfluoroethyl)trifluorophosphate ([C4mim][PF6] + [C4mim][NTf2], [C4mim][PF6] + [C4mim][FAP], [C4mim][FAP] + [C4mim][NTf2]) in a wide concentration range at 303.15 K and complemented it with quantum mechanical calculations. Three pure ionic liquids underwent the same kind of analysis for comparison purposes. We found that the addition of the [FAP]--anion to a mixture enhances the segregation of non-polar domains and weakens the hydrogen-bond network. The H-bonds in the studied mixtures are rather weak, as follows from QTAIM analysis, with the rarest occurrence for the [FAP]--anion. The competition of two anions in the mixtures for the most acidic hydrogen of the 1-butyl-3-methylimidazolium cation is reported. In most of the cases, the smaller anion ([PF6]- or [NTf2]-) with stronger charge concentration displaces the bigger one ([NTf2]- or [FAP]-) from the preferred coordination site. The existing nano-segregation in some mixtures notably slows down ion diffusion. Our results show that the differences in anion size, shape and nature are the main reasons for nano-segregation and the non-ideal behavior of ionic liquid mixtures.