In this paper we discuss two different techniques for calculating the $\mathbb Z_2$-cohomology cup-length -- one based on fiberings and a result of Horanska and Korbaš, and the other based on Gröbner bases. We use these techniques to obtain $\mathbb Z_2$-cohomology cup-length or bounds for the $\mathbb Z_2$-cohomology cup-length of some of the real flag manifolds $F(1,\dots1,2,\dots,2,n)$.