Perhaps I'm wrong but I thought this was ok by Bertini.

Choose a general hypersurface $H_1$ containing $X$ (ie, choose a general linear combination of the generators of the ideal of $X$, make sure this isn't a pencil). Choose another general hypersurface $H_2$ containing $X$. Repeat this process. Eventually we end up with an intersection of $n - \dim X$ hypersurfaces containing $X$. Call this reducible variety $Y$. This has only one irreducible component besides $X$ by Bertini's theorem (the base locus was $X$ and its not a pencil). Is this what you had in mind?

I believe I've seen this in work of Kawakita and also Ein-Mustata on singularities (see in particular, lci defect ideals).

Perhaps I'm wrong but I thought this was ok by Bertini.

Choose a general hypersurface $H_1$ containing $X$ (ie, choose a general linear combination of the generators of the ideal of $X$, make sure this isn't a pencil). Choose another general hypersurface $H_2$ containing $X$. Repeat this process. Eventually we end up with an intersection of $n - \dim X$ hypersurfaces containing $X$. Call this reducible variety $Y$. This has only one irreducible component besides $X$ by Bertini's theorem (the base locus was $X$ and its not a pencil). Is this what you had in mind?

I assume this must come up in linkage theory (discussed in Eisenbud's book). I believe I've seen this in work of Kawakita and also Ein-Mustata on singularities (see in particular, lci defect ideals).