Is a variety a local complete intersection if it is locally a complement of to a smooth $N$-dimensional affine of $N-m$ affine subvarieties?

Question

If an equidimensional variety $V$ of dimension $m$ is locally a set-theoretic complete intersection (i.e., it can be covered by open subvarieties of certain intersections of $N-n$ hypersurfaces in $P^N$ for a large enough $N$; we consider set-theoretic intersections here) then it can also be covered by complements to smooth affine varieties $A_i$ of dimension $N$ of unions of $N-m$ of open affine subvarieties of $A_i$. I wonder: is the converse true also? What can be said about the versions of these statements for $N-m+c$ instead of $N-m$ (for some $c>0$; cf. How would you call a variety that is locally a complete intersection up to defect c?)? I don't know whether the answers depend on the base field.