Let C⊂Pr be a general curve of genus g embedded via a general linear series of degree d. The Maximal Rank Conjecture asserts that the restriction maps H0(OPr(m))→H0(OC(m)) are of maximal rank; this determines the Hilbert function of C.In this paper, we prove an analogous statement for the union of hyperplane sections of general curves. More specifically, if H⊂Pr is a general hyperplane, and C1,C2,…,Cn are general curves, we show H0(OH(m))→H0(O(C1∪C2∪⋯∪Cn)∩H(m)) is of maximal rank, except for some counterexamples when m=2.As explained in [5], this result plays a key role in the author's proof of the Maximal Rank Conjecture [7].