A celebrated theorem by Martens states that the dimension of the locus of line bundles of fixed degree \(d\) with at least \(k\) sections on a smooth, projective, irreducible curve of genus \(g\) doesn't exceed \(d-2k+2\). We establish similar results for various Brill-Noether loci and give some applications for higher rank Brill-Noether loci.