  A vertex coloring of a graph $G$ is called distinguishing if no non-identity
automorphisms of $G$ can preserve it. The distinguishing number of $G$, denoted
by $D(G)$, is the minimum number of colors required for such a coloring, and
the distinguishing threshold of $G$, denoted by $\theta(G)$, is the minimum
number $k$ such that every $k$-coloring of $G$ is distinguishing. As an
alternative definition, $\theta(G)$ is one more than the maximum number of
cycles in the cycle decomposition of automorphisms of $G$. In this paper, we
characterize $\theta (G)$ when $G$ is disconnected. Afterwards, we prove that,
although for every positive integer $k\neq 2$ there are infinitely many graphs
whose distinguishing thresholds are equal to $k$, we have $\theta(G)=2$ if and
only if $\vert V(G)\vert =2$. Moreover, we show that if $\theta(G)=3$, then
either $G$ is isomorphic to one of the four graphs on~3 vertices or it is of
order $2p$, where $p\neq 3,5$ is a prime number. Furthermore, we prove that
$\theta(G)=D(G)$ if and only if $G$ is asymmetric, $K_n$ or $\overline{K_n}$.
Finally, we consider all generalized Johnson graphs, $J(n,k,i)$, which are the
graphs on all $k$-subsets of $\{1,\ldots , n\}$ where two vertices $A$ and $B$
are adjacent if $|A\cap B|=k-i$. After studying their automorphism groups and
distinguishing numbers, we calculate their distinguishing thresholds as
$\theta(J(n,k,i))={n\choose k} - {n-2\choose k-1}+1$, unless $ k=\frac{n}{2}$
and $i\in\{ \frac{k}{2} , k\}$ in which case we have
$\theta(J(n,k,i))={n\choose k}$.
