The short answer: no, and wearing bras more can help your health. The long answer is below.
In 1955, a book was published by researchers Sydney Ross Singer and Soma Grismaijer. Dressed to Kill was an (inaccurate) write-up that attempted to link breast cancer and wearing a bra. They proposed that wearing a bra, especially ones that fitted properly and not loosely or had underwire, especially for more than eight hours a day, drastically increased the likelihood of cancer.
I'm going to pause and let you look at the Wikipedia article linked in the title in the paragraph above. Go ahead. I'll still be here.
This book was completely false. Who says so?
The American Cancer Institute.
The American Cancer Society.
Dr. Jen Gunter, an OB/GYN who writes for USA Today.
The collective National Institutes of Health.
These aren't all of the critics. These are just the ones I found in the last five minutes because I'm lazy.
Dressed to Kill was the first instance of this theory that proliferated beyond old wives' tales. Ever since then, you can find this theory cropping back up on natural health blogs and books, in conversation, even from some doctors.
The issue is that not only was this a self-published book about a theory, the theory was repeatedly found to be completely false. The support for it was one study which was completely misinterpreted. If you go and look at the study cited in it, you will see the only conclusion they found evidence for was that women with larger breasts were at a higher risk for cancer because they had more breast tissue. There were no conclusions that were able to be drawn about bras. Length of bra wear was even more proven false. Constant support was found to give the best breast tissue health and allow it, the ligaments, the milk glands, and the supportive blood and lymph systems to thrive.
Here is a rough drawing of the lymph system of the breast. There are no nodes inside the breast itself, contrary to what Singer and Grismaijer claim. The only lymph system located on the underside of the breast is made of small capillaries that do not extend down onto the stomach, and the connections from the breast lymph system to the abdominal lymph system are located inside the ribcage, not along the skin under the breast.
However, there is plenty of evidence to support that more support could better support your breast health. (see what I did there?)
For instance, both breasts and testicles have a higher risk for cancer if they have endured some kind of physical trauma in the past five years. Some of these examples of trauma are common, everyday occurrences like excessive bouncing. There is also evidence that
How do you protect your body best? Well, for your balls, some nice supportive underwear will help, and a cup will protect during activities that are more risky. For breasts? Well...
Wear a bra.
Yep. Dressed to Kill literally recommends you do the opposite of what can help protect your breasts, lift them more, and avoid other breast issues. Not only that, but some of the recommendations that have spun off of this publication's theories are hurting you.
Cupless bras, bras without any padding, bras without underwire, bras that are made of only a thin piece of fabric? Only for sleeping, preferably not at all. The lack of support these give increases your risk for many health and cosmetic issues.
The best bra to protect your breasts is a padded bra that has good coverage. So those of you who swear by the push-up bra: Good on you! You are wearing the safest bra possible. Plenty of light padding on the front helps cushion any impact, and the plush padding along the bottom helps protect your sensitive breast tissue from everyday bouncing. It should be noted that theories that push-up bras can cause breast sag are also unsubstantiated. In fact, the best way to prevent sagging and the issues that can come with it (stretch marks, lower milk production, ect.) is to give constant proper support. Even padded bras with underwire that are not push-up are still good, although more padding is best. And don't think that a little bra that barely covers your nipples will do as much good as a full-coverage. Some sexiness is still allowed, but if you aren't trying to show much cleavage consider a much more enveloping bra for yourself.
Some other bra tips:
Wearing a bra for more than two days in a row stretches it and reduces it's elasticity and support. Repeating this will wear the bra out permanently, rendering it much less useful and irreparable. Throw out any bras that this happens to, so you will not be tempted to wear it and will be forced to wash and wear your good ones.
Wash in hot water, rinse in cold water. Hand-wash with some detergent that is delicate-friendly. I like to wash my wife's bras in the tub, so I can do up to five or six at a time.
I hope these help you, and help clear up the mis-information I've been hearing for years now.