We examine models for the physical conditions in the dust envelopes around the closest and most conspicuous examples of luminosity class III red giants with infrared excesses such as δ And. (1) It has been previously suggested that the dust is sporadically ejected from the stars, but for most such stars, this model seems unlikely. (2) Another possibility is that in some cases we might be witnessing emission from interstellar dust that happens to be near the star, a ``cirrus hot spot.'' Since 70% of the red giants with infrared excesses lie within 100 pc of the Galactic plane where this phenomenon must sometimes occur, many of the excesses might be explained by this effect. However, a difficulty with this model for at least a few bright sources is that if the clouds have a uniform density, we expect sizes at 60 µm that are about a factor of 10 larger than found for the best studied examples. It seems likely that some class III giants do possess circumstellar dust. (3) The inferred mass of dust around some class III red giants is larger than 1026 g, more matter than would be expected when a Vega-type star evolves off the main sequence. Because the dust is inferred to be more than 100 AU from the star, we hypothesize that the large inferred dust masses is the result of the disintegration of comets. This model can be tested by using the Space Infrared Telescope Facility to measure the 60 µm sizes.