In this question I claim that every nested sequence of bounded closed subsets of a metric space has nonempty intersection if and only if the space has the Heine-Borel property. However, there's something that can throw a wrench in the proof: what if it is possible for there to be an uncountable collection of closed subsets with empty intersection such that every countable subcollection has nonempty intersection?
Is this possible in a metric space?
Let $X$ be an uncountable set endowed with the discrete metric. Then the family $\{X\setminus\{x\}\,|\,x\in X\}$ is an uncountable family of closed subsets with empty intersection. But no countable subfamily has that property.
The property that "if a family of closed sets has an empty intersection, then there is a countable subfamily with empty intersection", has a name. It's called Lindelöf. In a metric space this is equivalent to having a countable dense subset (separable), and many other such countability properties.
Hence Santos' example was the standard example of a non-separable metric space.