For simplicity I want to stick to the compact, orientable case. Further i want to assume, that $N$ has the structure of a CW-complex.

It might still be interesting to consider the case, where the dimension of $M$ and the dimension of $N$ are different; for example $D^2\simeq pt$ and $[0;1]\times S^1\simeq S^1$.

To find the dimension of $M$ one has to consider the structure of the (co-)jhomology and the $\cap$-product. The dimension of $M$ is the dimension of the highest non-vanishing homology $H_*(N)$. Futhermore $H_*(N)$ and $H^*(N)$ have to satisfy Poincare duality, so that the highest non-vanishing homology has to be $\mathbb{Z}$ and if one picks a generator, the $\cap$-product with this generator has to give isomorphisms. 
If this is satisfied, $N$ is called a "Poincare complex". (One can consider some simple examples of manifolds with boundaries like a disk bundle over a manifold or $S^1\times S^1\setminus D^2$).

The question whether a given Poincare complex is homotopy equivalent to a manifold (the other way round) is one classical problem in surgery theory.

There is a involved obstruction process coming, which is very roughly described in the [wikipedia][1]. More details can be found for example in the books mentioned there.

There should also be twisted versions of Poincare duality int the nonorientable case, which should also give a complete answer.

I do not know, whether the additional assumption, that the Poincare complex $N$ is indeed a manifold with boundary has any consequences for the surgery obstructions.

  [1]: http://en.wikipedia.org/wiki/Surgery_theory#The_surgery_approach