Partitioning any language to regular?

One time I saw a theorem that says something like any language can be partitioned into regular languages, but I can't remember exactly what the theorem is.
Could somebody tell me, is this true? What is this theorem?

Every language can be partitioned into a countable union of singleton languages (languages of the form $\{w\}$). Since singleton languages are regular, we get that every language can be partitioned into infinitely many regular languages.

StackExchange Computer Science Q#49512, answer score: 5