For dimension 4 the question has not been settled – no‐one presently knows a necessary and sufficient condition that tells exactly when a compact topological 4–manifold is (homeomorphic to) a CW–complex. What is known can be summarized as follows.

1. The “easy” direction (already mentioned in the question).
   • If a 4–manifold is smoothable, Kirby showed that it possesses a finite handle decomposition.
   • A finite handle decomposition is, after forgetting the thickening of the handles, nothing but a finite CW–structure, so every smooth (hence every smoothable) compact 4–manifold is a finite CW–complex.

2. What happens when the manifold is not smoothable?
   • At present there is no example of a compact topological 4–manifold that is known not to admit any CW–structure.
   • Nor is there a theorem asserting that every compact topological 4–manifold does have such a structure. The question is completely open.

The most prominent nonsmoothable example, Freedman’s (E_{8})–manifold, illustrates the situation. Freedman constructs it by starting with a 0–handle and eight 2–handles whose “cores’’ are replaced by Casson handles; the result is compact but it is not known whether the space can be written as a CW–complex (finite or countable). In particular there is no argument that turns a Casson handle into a union of a countable collection of genuine cells whose attaching maps satisfy the CW axioms, and there is no argument that forbids such a reinterpretation either.

3. Relation with triangulations and higher dimensions.
   • In dimensions ≤3 every topological manifold is triangulable, hence is a (finite) CW–complex.
   • In dimensions ≥5 there are non-triangulable manifolds (due to the Kirby–Siebenmann and Galewski–Stern obstructions), but they can still be shown to be CW–complexes. One proves this by cutting the manifold into pieces which are products with intervals, giving them handle decompositions relative to their boundaries, and then re-assembling.
   • Dimension 4 is the only dimension where the standard arguments break down. The difficulty is precisely the replacement of 2–handles by Casson handles; no one knows whether those can always be converted into CW–cells or whether there are obstructions to doing so.

4. What would settle the problem?
   • A single example of a compact 4–manifold that provably fails to be a CW–complex would answer the question negatively.
   • A constructive way of turning every Casson handle into a true 2–cell (possibly together with some auxiliary 3- and 4-cells) would answer it positively.

In short

Smoothable ⇒ handle decomposition ⇒ finite CW–complex.

Nonsmoothable ? No general result; no known counterexample.

So at the present time the answer to “When is a compact topological 4–manifold a CW complex?” is: “Precisely when it is smoothable – in all other cases we do not know; it is an open problem whether every compact topological 4–manifold (for example Freedman’s (E_{8}) manifold) admits any CW–structure at all.”