In Nakahara's book "Geometry, Topology and Physics" (Ch. 8, about the almost complex structure) they write:
Note that any 2$m$-dimensional manifold locally admits a tensor field $J$ [type (1,1)] which squares to $-id_{2m}$ [on $T_pM$].
Now, in view of the definition of almost complex manifold (some pages below in the book):
$M$ is called an a.c.m. if there exists a $(1,1)$-type tensor field $J$ (called the almost complex structure) such that $J_p^2 = -id_{T_pM}$ at each point $p \in M$
it seems to me that the first quoted sentence says something like "every 2$m$-dimensional manifold is almost complex". But, this cannot be true, since $S^4$ is the counterexample.
Where am I wrong?
To extend a little bit Grigory's comment:
In geometry and topology you often deal with things that can be defined locally and not globally. Some examples that may already be more familiar to you
Non-vanishing vector fields. Let $M$ be any smooth $n$-manifold and let $U$ be a coordinate chart. Since $U\subset \mathbb{R}^n$ is an open subset, $\partial_{x^1}$ is a non-vanishing vector field on $U$. But we cannot always find a global non-vanishing vector field on $M$.
Orientation. Let $M$ be any smooth $n$-manifold and let $U$ be a coordinate chart. Since $U\subset\mathbb{R}^n$ is an open subset, it is orientable. But $M$ may not be orientable.
Analytic functions. Let $M$ be any compact Riemann surface. Locally there exists many complex analytic functions. Globally there are none.