Within a generalized Caldeira-Leggett model, we analyze the conditions under which a bosonic heat bath can entangle two microscopic quantum systems at a distance r. We find that the attainable entanglement is extremely distance-sensitive. Significant entanglement can only be achieved if the systems are within a microscopic distance that is of order of the cutoff wavelength lambda of the system-bath interaction. At larger distances, the maximal entanglement is exponentially suppressed with a decay length of order lambda. We conclude that entanglement generation via a heat bath is not suitable for entangling remote objects.