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.*][1]
Proc. Amer. Math. Soc. 13 1962 763–764. 

$\mathbb{RP}^9$ does not embed into $\mathbb{R}^{16}$.


  [1]: https://www.ams.org/journals/proc/1962-013-05/S0002-9939-1962-0143222-6/S0002-9939-1962-0143222-6.pdf