We study close approaches in hierarchical triple systems (a binary system with a tertiary), motivated by a new model for type Ia supernovae involving direct collisions of white dwarfs (WDs) in triple systems. We show that the ability of the inner binary to achieve very close approaches (where the distance $r/a$ shrinks to $\leq 10^{-5}$ of its original size) can be analytically predicted from initial conditions, as confirmed by full N-body simulations of such systems. We find that the rate of close approaches is roughly linear with the mass of the tertiary. The chance for clean collisions is $\sim1$\% in triple systems with WD-WD inner binaries and typical M-dwarf tertiaries. This sets challenging constraints on the collisional model in triple systems.