In HTT, 5.5.6.19, the possible ambiguity of the notation $\tau_{\leq k} \mathcal C$ is discussed. By the first definition, it denotes the subcategory on the $k$-truncated objects but it can also be read as the essential image of $\mathcal C$ under the truncation functor. The cited proposition states that there is no problem because those subcategories coincide.
But there is another way, how to view $\tau_{\leq k}\mathcal C$, namely as the value of the truncation functor on $\mathrm{Cat}_\infty$, the $\infty$-category of $\infty$-categories. Maybe I am just missing something obvious, but I am wondering, does this also yield the same?
An $\infty$-category being $0$-connected means that its morphism spaces are $(-1)$-connected, that is, they are homotopically subsingletons. If it is the nerve of a $1$-category, it must therefore be a poset. Now one counterexample is the $\infty$-category of spaces for which its subcategory of $0$-truncated objects is equivalent to the category of sets. But $\mathrm{Set}$ is not $0$-truncated in $\mathrm{Cat}_\infty$ since it is not equivalent to a poset.