We give a short argument that yields a new lower bound on the number of subsampled rows from a bounded, orthonormal matrix necessary to form a matrix with the restricted isometry property. We show that a matrix formed by uniformly subsampling rows of an N×N Hadamard matrix contains a K-sparse vector in the kernel, unless the number of subsampled rows is Ω(KlogKlog(N/K)) --- our lower bound applies whenever min(K,N/K)>logCN. Containing a sparse vector in the kernel precludes not only the restricted isometry property, but more generally the application of those matrices for uniform sparse recovery.