Manifold embedded in $R^{n+1}$ with a submanifold that doesn't embed in $R^n$

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

Here's another way to get examples, in codimension one and in low dimensions. There are lots of oriented closed 3-manifolds that don't embed in 4-space, for example any 3-manifold $M$ with $H_1(M) \cong \mathbb{Z}_{2n}$. But a theorem of Hirsch says that $M$ embeds in $\mathbb{R}^5$. By a standard transversality argument (like the one that produces Seifert surfaces for higher-dimensional knots) $M = \partial W$ for some oriented $W \subset \mathbb{R}^5$. Now $W$ has a trivial normal bundle, so $W \times I \subset \mathbb{R}^5$, and so $M \subset \partial(W \times I)$, which is just the double of $W$.

You can do a similar thing with an oriented closed $n$-manifold that doesn't embed in $n+1$-space but does embed in $n+2$ space. These aren't so easy to come by; for $n=4$ they were constructed by Tim Cochran (Inventiones Math. 77 (1984), 173--184). If you don't mind non-orientable examples, then you can do this with $n=2$. For a Klein bottle embeds in the quaternionic space form $S^3/Q8$, which in turn embeds in $\mathbb{R}^4$. (It is the boundary of a tubular neighborhood of an embedded $RP^2$.) But a Klein bottle cannot embed in $\mathbb{R}^3$.

If you are interested in an example in codimension 2 examples (which also happens to involve two orientable manifolds), according to the table on page 2 in the survey

Davis, Donald M. Embeddings of real projective spaces, Bol. Soc. Mat. Mex. 4 (1998) 115-122.

$\mathbb{RP}^{39}$ embeds in $\mathbb{R}^{71}$, but $\mathbb{RP}^{37}$ does not embed in $\mathbb{R}^{70}$.

The first result is credited to:

Rees, Elmer, Embeddings of real projective spaces, Topology 10 (1971) 309-312.

and the second to

Adem, José, Gitler, Samuel, and Mahowald, Mark E. Embedding and immersion of projective spaces, Bol. Soc. Mat. Mex. 10 (1965) 84-88.