One approach to a homeomorphism classification of a closed manifolds simply homotopy equivalent to a closed manifold $X$ of dimension $>4$ is to compute the topological structure set $\mathcal S^s_\text{TOP}(X)$ and the group of homotopy classes of simple homotopy equivalence $\text{Aut}_s(X)$. Then $\text{Aut}_s(X)$ acts on $\mathcal S^s_\text{TOP}(X)$ be composition and the quotient is the desired set of homeomorphism classes closed manifolds simply homotopy equivalent to $X$.

For $X=\mathbb RP^n$ the topological structure set is <a href="http://www.map.mpim-bonn.mpg.de/Fake_real_projective_spaces#Homeomorphism_classification_of_topological_actions">known</a>, and the structure set always has at least 4 elements, and it is finite unless $n-3$ is divisible by $4$.
 
The group $\text{Aut}_s(\mathbb RP^n)$ is trivial if $n$ is even and has order $2$ if $n$ is odd. See <a href="http://www.numdam.org/item/CM_1973__26_2_119_0.pdf"> Corollary 6</a> in "Coverings of fibrations" by Becker and Gottlieb.

Thus there are lots of fake even-dimensional real projective spaces in  dimensions $>4$. 

By Freedman's work this can be extended to dimension $4$, see  <a href="https://core.ac.uk/download/pdf/82800539.pdf"> Invariant knots of free involutions on $S^4$</a> by Ruberman for examples of fake $\mathbb RP^4$.