If I have understood the table at
http://www.lehigh.edu/~dmd1/immtable
correctly, then $\mathbb{RP}^{10}$ embeds into $\mathbb{R}^{17}$. But by
Mahowald, Mark On the embeddability of the real projective spaces. Proc. Amer. Math. Soc. 13 1962 763–764.
$\mathbb{RP}^9$ does not embed into $\mathbb{R}^{16}$.
