Let ℓ denote a Banach sequence space with a monotone norm in which the canonical system (e n) n is an unconditional basis. We show that the existence of an unbounded continuous linear operator T between ℓ-Köthe spaces λ ℓ (A) and λ ℓ (C) which factors through a third ℓ-Köthe space λ ℓ (B) causes the existence of an unbounded continuous quasidiagonal operator from λ ℓ (A) into λ ℓ (C) factoring through λ ℓ (B) as a product of two continuous quasidiagonal operators. Using this result, we study when the triple (λ ℓ (A), λ ℓ (B), λ ℓ (C)) satisfies the bounded factorization property BF (which means that all continuous linear operators from λ ℓ (A) into λ ℓ (C) factoring through λ ℓ (B) are bounded). As another application, we observe that the existence of an unbounded factorized operator for a triple of ℓ-Köthe spaces, under some additional assumptions, causes the existence of a common basic subspace at least for two of the spaces.