We discuss the question of geometric formality for rationally elliptic manifolds of dimension $6$ and $7$. We prove that a geometrically formal six-dimensional biquotient with $b_2=3$ has the real cohomology of a symmetric space. We also show that a rationally hyperbolic six-dimensional manifold with $b_2\leq 2$ and $b_3=0$ can not be geometrically formal. As it follows from their real homotopy classification, the seven-dimensional geometrically formal rationally elliptic manifolds have the real cohomology of symmetric spaces as well.