There are actually counterexamples in real dimension $4$.

The first examples of compact almost-complex $4$-manifolds not admitting any complex structure were produced by Van de Ven in his paper "On the Chern numbers of some complex and almost-complex manifolds".

In fact, he obtained restrictions on the Chern numbers of an algebraic surface and constructed some almost-complex $4$-manifolds violating them, hence showing that no almost complex structure in these examples could be integrable.

Later on, Brotherton constructed some counterexamples which are also parallelizable, see this paper.

There are actually counterexamples in real dimension $4$.

The first examples of compact almost complex $4$-manifolds admitting no complex structure were produced by Van de Ven in his paper "On the Chern numbers of some complex and almost-complex manifolds".

In fact, he obtained restrictions on the Chern numbers of an algebraic surface and constructed some almost complex $4$-manifolds violating them, hence showing that no almost complex structure in these examples could be integrable.

Later on, Brotherton constructed some counterexamples with trivial tangent bundle, see the article "Some parallelizable 4-manifolds not admitting a complex structure".

There are actually counterexamples in real dimension $4$.

The first examples of compact almost-complex $4$-manifolds not admitting any complex structure were produced by Van de Ven in his paper "On the Chern numbers of some complex and almost-complex manifolds". In fact, he obtained restrictions on the Chern numbers of algebraic surfaces and constructed some almost-complex $4$-manifolds violating them, hence showing that the almost complex structures in these example could not be integrable.

Later on, Brotherton constructed some counterexample which are also parallelizable, see this paper.

There are actually counterexamples in real dimension $4$.

The first examples of compact almost-complex $4$-manifolds not admitting any complex structure were produced by Van de Ven in his paper "On the Chern numbers of some complex and almost-complex manifolds". 

In fact, he obtained restrictions on the Chern numbers of an algebraic surface and constructed some almost-complex $4$-manifolds violating them, hence showing that no almost complex structure in these examples could be integrable.

Later on, Brotherton constructed some counterexamples which are also parallelizable, see this paper.