How “disconnected” can a continuum be? A continuum is a compact connected metrizable topological space.
Given a cardinal $\kappa$, a topological space $X$ is called $\kappa$-connected if it is not possible to write $X$ as the disjoint union of more than one and at most $\kappa$ many closed subsets. In particular a space is connected iff it is $2$-connected iff it is $n$-connected for all $2\leq n<\aleph_0$.
It is a standard result that continua are $\aleph_0$-connected and, by writing a continuum as a union of singletons, it is clear that continua are $\mathfrak c$-disconnected.
Given a continuum $X$ let $\mathrm{disc}(X)$ denote the smallest $\kappa$ such that $X$ is $\kappa$-disconnected.
Is it consistent with $\mathsf{ZFC}$ to have a continuum $X$ with $\aleph_0<\mathrm{disc}(X)<\mathfrak c$?
Is it a theorem of $\mathsf{ZFC}$ that if $X$ and $Y$ are nontrivial continua, then $\mathrm{disc}(X)=\mathrm{disc}(Y)$?
Assuming a positive answer to the previous two questions, is $\mathrm{disc}(X)$ equal to some standard cardinal invariant of the continuum?
 A: The answer to all three of your questions is yes.
The cardinal $\mathrm{disc}([0,1])$ is discussed in this MO question of Taras Banakh. He calls this cardinal the Sierpiński cardinal and denotes it $\acute{\mathfrak n}$. The comments and answers there give a good bit of information about this cardinal, including that it is consistent to have $\acute{\mathfrak n} < \mathfrak{c}$. This consistency result was first proved by Stern, who showed that it is true in the random real model.
The Sierpiński cardinal also goes by another name in the literature: $\mathfrak{a}_T$. This is because it can be characterized as the smallest size of an infinite maximal family of almost disjoint subtrees of the infinite binary tree $2^{<\omega}$.
Arnie Miller shows in Theorem 3 of this paper that $\mathfrak{a}_T = \aleph_1$ if and only if every uncountable Polish space can be partitioned into $\aleph_1$ closed sets. As every continuum is Polish, this shows that $\mathrm{disc}(X) = \mathfrak{a}_T = \aleph_1$ for every continuum $X$, provided that $\mathfrak{a}_T = \aleph_1$. This almost answers your second question.
Miller's argument does not readily generalize to cardinals $>\!\aleph_1$. Recently I wrote a paper about partitions of Polish spaces into closed sets (and Borel sets, more generally), and I found a a way to extend Miller's result to cardinals $>\!\aleph_1$ (via a different approach): see Theorem 2.4 here. So it is true that $\mathrm{disc}(X) = \mathfrak{a}_T$ for any continuum $X$; and more generally, $\mathfrak{a}_T$ is equal to the smallest size of any partition of a Polish space $X$ into uncountably many closed sets (or compact sets, or $F_\sigma$'s).
