We consider a rate-distortion problem with side information at multiple decoders. Several upper and lower bounds have been proposed for this general problem or special cases of it. We provide an upper bound for general instances of this problem, which takes the form of a linear program, by utilizing random binning and simultaneous decoding techniques and compare it with the existing bounds. We also provide a lower bound for the general problem, which was inspired by a linear-programming lower bound for index coding, and show that it subsumes most of the lower bounds in literature. Using these upper and lower bounds, we explicitly characterize the rate-distortion function of a problem that can be seen as a Gaussian analogue of the "odd-cycle" index coding problem.