We investigate the problem of introducing consistent self-couplings in free theoriesfor mixed tensor gauge fields whose symmetry properties are characterized byYoung diagrams made of two columns of arbitrary (but different) lengths. We provethat, in flat space, these theories admit no local, Poincaré-invariant, smooth, selfinteractingdeformation with at most two derivatives in the Lagrangian. Relaxingthe derivative and Lorentz-invariance assumptions, there still is no deformation thatmodifies the gauge algebra, and in most cases no deformation that alters the gaugetransformations. Our approach is based on a Becchi-Rouet-Stora-Tyutin (BRST)-cohomology deformation procedure"