Self-avoiding lattice walks are often used as minimalist models of proteins. Typically, the polypeptide chain is represented as a lattice walk with each amino acid residue lying on a lattice point, and the Hamiltonian being a sum of interactions between pairs of sequentially nonadjacent residues on adjacent points. Interactions depend on the types of the two residues, and there are usually two or more types. A sequence is said to fold to a particular "native" conformation if the ground state is nondegenerate, i.e., that native conformation is the unique global energy minimum conformation. However, real proteins have some flexibility in the folded state. If this is permitted in a lattice model, the most stably and cooperatively folding sequences have very disordered native states unless the Hamiltonian either favors only a few specific interactions or includes a solvation term. The result points the way toward qualitatively more realistic lattice models for protein folding.