A (special case of a) theorem of Gromov says for any $n\in \mathbb{N}$ there exists a constant $C(n)$ such that for any smooth connected closed $n$-dimensional Riemannian manifold with non-negative sectional curvature the sum of all of its Betti numbers is at most $C(n)$.


On the other hand, for $n=2$ only sphere, torus, real projective plane, and (possibly- I do not know) connected sum of 2 real projective planes, can carry a non-negatively curved metric. (This follows from the [Gauss-Bonnet theorem](https://en.wikipedia.org/wiki/Gauss%E2%80%93Bonnet_theorem) and [topological classification of surfaces](https://en.wikipedia.org/wiki/Surface_(topology)) with non-negative Euler characteristic.) 



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> What happens when $n>2$? For which $n>2$ it is known that there are infinitely many homeomorphism/ homotopy types of $n$-manifolds which can carry a Riemannian metric with non-negative sectional curvature?
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Homeomorphism/ homotopy types of non-negatively curved manifolds
As mentioned in comments, the first dimension where an infinite family of pairwise non-homeomorphic closed nonnegatively curved manifolds occurs is $3$ (the lens spaces). The question becomes more challenging for simply-connected manifolds. If (as expected) simply-connected nonnegatively curved manifolds are rationally elliptic, then the first infinite family occurs in dimension 6, see [Curvature, diameter, and quotient manifolds](https://www.math.ucla.edu/~totaro/papers/public_html/curv.pdf) by
B. Totaro. In general, the question is well-studied and there are many examples (easily found by searching online).


