Bipolynomial Hilbert functions

Let X⊂Pn be a closed subscheme and let HF(X,⋅) and hp(X,⋅) denote, respectively, the Hilbert function and the Hilbert polynomial of X. We say that X has bipolynomial Hilbert function if HF(X,d)=min{hp(Pn,d),hp(X,d)} for every d∈N. We show that if X consists of a plane and generic lines, then X has bipolynomial Hilbert function. We also conjecture that generic configurations of non-intersecting linear spaces have bipolynomial Hilbert function.