The Steenrod operations on mod 2 cohomology imply the vanishing of some characteristic numbers. Specifically, if $p(w_1,w_2,\ldots)\in H^k(M^n;Z/2)$ for $k\lt n$ then $0=\langle \sum_{i+j=n-k}w_i{\rm Sq}^{j} p, [M^n]\rangle$. Thom showed that *all* relations between characteristic numbers arise in this way. This allowed him to compute the bordism ring of unoriented manifolds exactly: it is $Z/2[m_k\vert k$ not of the form $2^j-1]$. This gives you the tight bound you were asking for.

But, as others have already pointed out, there are plenty of good books & papers on this, so you should probably just dive into the library.