  Let $(G_n)$ be a sequence of finite connected vertex-transitive graphs with
volume tending to infinity. We say that a sequence of parameters $(p_n)$ is a
percolation threshold if for every $\varepsilon > 0$, the proportion
$\left\lVert K_1 \right\rVert$ of vertices contained in the largest cluster
under bond percolation $\mathbb{P}_p^G$ satisfies both \[ \begin{split}
  \lim_{n \to \infty} \mathbb{P}_{(1+\varepsilon)p_n}^{G_n} \left( \left\lVert
K_1 \right\rVert \geq \alpha \right) &= 1 \quad \text{for some $\alpha > 0$,
and}
  \lim_{n \to \infty} \mathbb{P}_{(1-\varepsilon)p_n}^{G_n} \left( \left\lVert
K_1 \right\rVert \geq \alpha \right) &= 0 \quad \text{for all $\alpha > 0$}.
\end{split}\] We prove that $(G_n)$ has a percolation threshold if and only if
$(G_n)$ does not contain a particular infinite collection of pathological
subsequences of dense graphs. Our argument uses an adaptation of Vanneuville's
new proof of the sharpness of the phase transition for infinite graphs via
couplings [Van22] together with our recent work with Hutchcroft on the
uniqueness of the giant cluster [EH21].
