Drawing inspiration from a well-known conjecture of Chv\'atal on a toughness threshold guaranteeing graph Hamiltonicity, we investigate toughness properties of so-called arbitrarily partitionable (AP) graphs, which are those graphs that can be partitioned into arbitrarily many connected graphs with arbitrary orders, and can be perceived as a weakening of Hamiltonian and traceable graphs. In particular, we provide constructions of non-AP graphs with toughness about $\frac{5}{4}$, \textit{i.e.}, in which, when removing the vertices of any cut-set $S$, the number of resulting connected components is at most about $\frac{4}{5} |S|$. We also consider side related questions on graphs that can be partitioned arbitrarily into only a few connected graphs (with arbitrary orders). Among other things, we prove that not all $1$-tough graphs can always be partitioned into four connected graphs this way. As going along, we also raise several other questions and problems of interest on the topic.