  Over an infinite field $K$ with $\mathrm{char}(K)\neq 2,3$, we investigate
smoothable Gorenstein $K$-points in a punctual Hilbert scheme from a new point
of view, which is based on properties of double-generic initial ideals and of
marked schemes. We obtain the following results: (i) points defined by graded
Gorenstein $K$-algebras with Hilbert function $(1,7,7,1)$ are smoothable, in
the further hypothesis that $K$ is algebraically closed; (ii) the Hilbert
scheme $\mathrm{Hilb}_{16}^7$ has at least three irreducible components. The
properties of marked schemes give us a simple method to compute the Zariski
tangent space to a Hilbert scheme at a given $K$-point, which is very useful in
this context. Over an algebraically closed field of characteristic $0$, we also
test our tools to find the already known result that points defined by graded
Gorenstein $K$-algebras with Hilbert function $(1,5,5,1)$ are smoothable. In
characteristic zero, all the results about smoothable points also hold for
local Artin Gorenstein $K$-algebras.
