In every dimension $\ge 4$ there is a fake real projective space, i.e., a manifold that is homotopy equivalent but not homeomorphic to $RP^n$.
Here you can find a computation for the topological surgery structure set of $RP^n$. It is stated for $n>4$ but I think it extends to $n=4$ because $\mathbb Z_2$ is good in the sense of Freedman. To get a homeomorphism classification of homotopy $RP^n$'s we also need to know the group of homotopy self equivalences of (any manifold homotopy equivalent to) $RP^n$: it is $\mathbb Z_2$ if $n$ is odd and trivial if $n$ is even (see e.g. p.61 of Rutter's survey ``Spaces of homotopy self-equivalences''). Moreover, if $n$ is odd, then $RP^n$ admits a self-map of degree $-1$, namely, the map induced by reflection in an equator of $S^n$. Thus any homotopy self-equivalence of $RP^n$ is homotopic to a diffeomorphism.
Let $f_i: M_i\to RP^n$ be homotopy equivalences representing different elements in the structure set. If $d_{ij}: M_i\to M_j$ is a homeomorphism, then $f_i^{-1}f_j d_{ij}$ is a homotopy self-equivalence of $M_i$. By the previous paragraph $f_i^{-1}f_j d_{ij}$ is homotopic to a homeomorphism, so $f_i$, $f_j$ represent the same element in the structure set. Thus in every dimension there is a fake $RP^n$.
A word of caution: the above argument does not show that the fake $RP^n$ is smoothable. For example, the only fake $RP^4$ is not smoothable as I learned from ``Invariant knots of free involutions of $S^4$'' by Ruberman, see here. There do exist many smoothable fake $RP^n$'s, see e.g., Smooth free involutions on homotopy $4k$-spheres by Fintushel-Stern.