A mean-field lattice model is used to investigate some possible aspects of the low-temperature behavior of a lower critical solution temperature (LCST) polymer blend made miscible by a dilute distribution of a hydrogen-bonding comonomer. The nature of the binding interaction is examined for both a random comonomer distribution and for the case of identically end-modified chains. For the latter case, competition between the interaction responsible for the immiscibility of the homopolymer components and the attraction due to hydrogen bonding leads to a stable intermediate phase. For random distributions, polydispersity in the binding interaction restricts this intermediate phase to an extremely narrow window in reduced temperature, consistent with a "smeared" transition. The free-energy landscape contains local minima separated by energy barriers.