In this note we show that weakly compact operators from a Banach space $X$ into a complete (LB)-space $E$ need not factorize through a reflexive Banach space. If $E$ is a Fréchet space, then weakly compact operators from a Banach space $X$ into $E$ factorize through a reflexive Banach space. The factorization of operators from a Fréchet or a complete (LB)-space into a Banach space mapping bounded sets into relatively weakly compact sets is also investigated.