The likelihood that an undercooled liquid vitrifies or crystallizes depends on the cooling rate R. The critical cooling rate R(c), below which the liquid crystallizes upon cooling, characterizes the glass-forming ability (GFA) of the system. While pure metals are typically poor glass formers with R(c)>10(12)K/s, specific multi-component alloys can form bulk metallic glasses (BMGs) even at cooling rates below R∼1 K/s. Conventional wisdom asserts that metal alloys with three or more components are better glass formers (with smaller R(c)) than binary alloys. However, there is currently no theoretical framework that provides quantitative predictions for R(c) for multi-component alloys. In this manuscript, we perform simulations of ternary hard-sphere systems, which have been shown to be accurate models for the glass-forming ability of BMGs, to understand the roles of geometric frustration and demixing in determining R(c). Specifically, we compress ternary hard sphere mixtures into jammed packings and measure the critical compression rate, below which the system crystallizes, as a function of the diameter ratios σ(B)/σ(A) and σ(C)/σ(A) and number fractions x(A), x(B), and x(C). We find two distinct regimes for the GFA in parameter space for ternary hard spheres. When the diameter ratios are close to 1, such that the largest (A) and smallest (C) species are well-mixed, the GFA of ternary systems is no better than that of the optimal binary glass former. However, when σ(C)/σ(A) ≲ 0.8 is below the demixing threshold for binary systems, adding a third component B with σ(C) < σ(B) < σ(A) increases the GFA of the system by preventing demixing of A and C. Analysis of the available data from experimental studies indicates that most ternary BMGs are below the binary demixing threshold with σ(C)/σ(A) < 0.8.