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 $\mathbb{R}^{16}$.