Let \({\mathcal P}={\mathcal P}_ 0\cup {\mathcal P}_ 1\) where \({\mathcal P}_ 0\) and \({\mathcal P}_ 1\) denote null and alternative hypotheses, and let \(C_{n,\alpha}\) be the critical region of an \(\alpha\)-level test for testing \({\mathcal P}_ 0\) versus \({\mathcal P}_ 1\) when a sample of size n is used. A sequence of tests \(\{C_{n,\alpha_ n}\}\) is said to be completely consistent if the associated risk \(R(C_{n,\alpha_ n},P)\to 0\) as \(n\to \infty\) for all \(P\in {\mathcal P}\). Under suitable conditions the author proves some connections between completely consistent tests, consistent estimators, and consistent sequences of tests.