Here is the answer to the question, kindly explained to me by Burt Totaro.

EDITED. This is an OPEN PROBLEM.

0) Apparently in the case of $\mathbb CP^n$ existence of a complex bundle without holomorphic structure is still an OPEN PROBLEM. Though it is believed that there should be plenty of examples starting from $n\ge 5$, coming from topologically indecomposable rank two bundles, apparently no such bundle was proven to be non-holomorphic as for today.

1) A topologically non-trivial rank 2 complex bundle with $c_1=0$, $c_2=0$ was constructed in

Rees, Elmer, Some rank two bundles on ${\rm P}_{n}\mathbb C$, whose Chern classes vanish. Variétés analytiques compactes (Colloq., Nice, 1977).

It was also claimed in this article that this bundle does not admit a holomorphic structure. But this claim was deduced from an article that contained a gap. So it is yet unknown if this particular bundle has holomorphic structure or not. This is discussed in
M. Schneider. Holomorphic vector bundles on ${\rm P}^n$. Seminaire Bourbaki
1978/79, expose 530.

This is why Okonek and Schneider write in their book p. 137 that this is an open problem.

2) On the positive side it is proven that every complex vector bundle on a smooth projective rational 3-fold has an holomorphic structure.

C. Banica and M. Putinar. On complex vector bundles on projective
threefolds. Invent. Math. 88 (1987), 427-438.

3) If one wants to construct examples of bundles on projective manifolds that are not necessarily Fanos it is possible to use the fact that the integral Hodge conjecture fails. Namely there are elements in $H^{2p}(X,\mathbb Z)$ which are in $H^{p,p}$
but which are not represented by an algebraic cycle. Kollar gave
such examples with $dim(X)=3$. A recent reference, which refers back to earlier results, is:

C. Soule and C. Voisin. Torsion cohomology classes and algebraic cycles
on complex projective manifolds. Adv. Math. 198 (2005), 107-127

4) One reason to expect examples of such bundles in higher dimensions is
Schwarzenberger's conjecture that every rank-2 algebraic vector bundle $E$,
on $\mathbb CP^n$ with $n\ge 5$ is a direct sum of two line bundles. So, for example,
if $c_1(E)=0$ then $c_2(E)=-d^2$ for some integer $d$, according to the
conjecture.