We investigate stability and robustness properties of a family of algorithms used to "coarsely quantize" band-limited functions. The algorithms we will consider are one-bit second-order sigma-delta quantization schemes and some modified versions of these. We prove that there exists a bounder region that remains invariant under the two-dimensional piecewise-affine discrete dynamical system associated with each of these quantizers. Moreover, this bounded region can be constructed so that it is robust under small changes in the quantizer. We will also show some interesting properties of the resulting binary sequences.