Manneville and Dubois‐Violette’s linear stability analysis of the roll‐cell instability in nematics is extended to account for the effect of finite rates of roll‐cell growth. The extended theory is applied to shearing flows of tumbling liquid crystalline polymers, in which the Ericksen number is typically a decade or more above the critical value for formation of the roll cells. It is found that, starting from a monodomain with director oriented perpendicular to both the flow and the flow gradient, at asymptotically high Ericksen numbers a wide band of wave numbers is unstable and the widthlmof the fastest growing roll cells is proportional to γ̇−1/4D1/2, where γ̇ is the shear rate andDis the gap between the shearing surfaces. This prediction of thin roll cells whose width depends only weakly on shear rate is consistent, in part, with roll‐cell observations of Srinivasarao and Berry for poly(1,4‐phenylene‐2,6‐benzobisthiazole), and with characteristic texture spacings measured in sheared poly(γ‐benzyl‐glutamate) solutions. These results suggest that the roll‐cell instability may provide a mechanism for texture refinement and director turbulence in shearing flow of tumbling nematics.