We study inclusive processes involving two heavy quarkonia in nonrelativistic QCD (NRQCD) and demonstrate that, in the presence of two P-wave Fock states, NRQCD factorization breaks down, leaving uncanceled infrared singularities. As phenomenologically important examples, we consider the decay Upsilon -> chi(cJ) + X via b (b) over bar(P-3(Jb){[}8]) -> c (c) over bar(P-3(J){[}1]) + gg and the production process e(+)e(-) -> J/psi + chi(cJ) + X via e(+)e(-) -> c (c) over bar(P-3(J){[}8]) +-> c (c) over bar(P-3(J){[}1])+ g. We infer that such singularities will appear for double quarkonium hadroproduction at next-to-leading order. As a solution to this problem, we introduce to NRQCD effective field theory new types of operators whose quantum corrections absorb these singularities.