We present a comprehensive computational and experimental examination of the Cr1-xVxO2 (0 <= x <= 0.5) system. The entire series crystallizes in the rutile structure, but the compounds exhibit significantly different magnetic properties depending on x. Lattice parameter a increases linearly with x, but the c parameter is slightly reduced due to vanadium-vanadium bonding. The V-for-Cr substitution creates Cr3+-V5+ pairs; this leads to competition between ferromagnetic (Cr4+-Cr4+) and antiferromagnetic (Cr3+-Cr3+) interactions such that the materials change from ferromagnetic to antiferromagnetic with increasing x. Weak ferromagnetic interactions arising from Cr4+ are observed even in the seemingly antiferromagnetic phases with the exception of x = 0.5, which contains only Cr3+. Density functional theory calculations are performed, but they incorrectly predict the x = 0.5 phase to be a half-metal. This is caused by an incorrect prediction of the oxidation states of chromium and vanadium.