It is easy to show that if the union of a finite family $\mathcal{C}$ of (more than one) connected sets is connected, then for any $C \in \mathcal{C}$ there must always be some other $C' \in \mathcal{C}$ such that $C \cup C'$ is connected. In fact I think this works if $\mathcal{C}$ is a locally finite collection.
But what if $\mathcal{C}$ is not a locally finite collection? I suspect that such $C'$ may not exist in general, but I cannot find a counterexample.
What about $\mathcal{C} = \{[\frac{1}{n}, 1]\mid n\in\mathbb{N}\}\cup\{\{0\}\}$?