I am going to convert my comments to an answer, since this seems to be a bit of a tough nut to crack.
First of all, the CNN article links to an article on Health.com which states the same thing. The only reference is that a Dr. Roshini Rajapaksa makes this assertion. There is no links or discussion of any studies or even any particular expertise in this. Dr. Roshini Rajapaksa is actually a Gastroenterologist, and as such may not be fully qualified to speak on the subject of breast tissue, or breast sagging. At her NYU web page, it shows that she has no studies listed to her credit. She does have numerous articles and publications in the field of Gastroenterology & Hepatology. From 1999 to 2008 (on her web page), all the publications are in her area of expertise. Later on though, in 2008, she started publishing in the Health Magazine, and the articles seem to step outside the bounds of her expertise.
As to the claim of breast sagging, I got led around to quite a few sites on the internet, and can't seem to find a great deal of impartial information on the subject. I examined first of all if there is actually a ligament involved in the physiology. On that count, there is. It is called the Cooper's ligament. This wikipedia article says:
Many women have held the mistaken belief that sagging is caused by the failure of the Cooper's ligaments to support the breast tissue. In fact, sagging is partly determined by genetic factors, but cigarette smoking, a woman's body mass index, her number of pregnancies, the size of her breasts before pregnancy, and age are all influencing factors.
The citation wikipedia gives is somewhat questionable in my opinion, because it seems to suffer from the same problem that the assertion that your sleep position can cause sagging.... That is, it doesn't cite any original research on the subject.
The medical term for sagging breasts is called Ptosis of the breast (as opposed to plain Ptosis that generally refers to eyelid drooping). Again, I am not able to locate any specific studies on breast ptosis and sleeping position. All the studies indicate that it's simply the combined effect of gravity, combined with time. With time being the main contributor, because ligaments just naturally wear out over time.
Aging: Because of the way that breasts are built, sagging is common and largely unavoidable. After all, breasts contain no muscles of their own to hold them up. Rather, breasts are made of fat, glands, milk ducts, and a type of connective tissue called the "Cooper's ligaments," which are thought to provide some structural support. As a woman gets older, these ligaments may get stretched and lose elasticity. Breasts may lose fullness and size as they lose their underlying support system of tissue and fat.
An advice column at Columbia University does say that:
Contrary to popular belief, going braless doesn't mean that your breasts are destined to droop. There's also nothing unhealthy about wearing one as long as it fits properly. Bras do not preserve the shape or perkiness of breasts. Sagging results from a number of factors unrelated to brassiere wearing, such as:
breast fat and tissue composition
softening of breast ligaments and loss of breast tissue with age
enlarged breasts during pregnancy, which causes the skin to stretch and remain stretched even after they revert to pre-pregnancy size
effects of gravity over a woman's lifetime
Again, the main factor appears to be age and time, and not sleeping position.
In all these cases, there is a mention of gravity. Physiologically, all the previously cited places indicate that if a woman has large breast, they will naturally sag more just because they are large. However, the Cooper's ligament still is able to accommodate that weight as part of its job. The only thing that affects the ligament tissue, from what I read in the papers, was the loss of elasticity due to natural aging. They will not cause accelerated sagging due to sleeping position (or not wearing a bra).
