In a word: no.
The "bras = cancer" myth originated with a book published in 1996 by Sydney Ross Singer and Soma Grismaijer called "Dressed to Kill" wherein the authors "conducted" a "study" and concluded that bras increased the risk of cancer several times over.
This was NOT a peer-reviewed study published in a journal. They simply surveyed women (without using blinding methods; this means they could ask leading questions), neglected to control for other factors known to correlate with breast cancer and published their results in a book, not a journal. Many ignorant folk read it, didn't recognize the unscientific nature of what they did, and took it to be truth. Even worse than that: they go on to develop a hypothesis for their results that, to a layperson, sounds plausible. (That bras restrict lymphatic circulation which lets toxins build up in the breast)
One of the factors they neglected to control for is the known correlation between higher BMI and higher rates of breast cancer - as Lola of Stacked said in the post that made me look into this idea, women with larger breasts are more likely to wear bras for longer periods of time. Women with larger breasts are also more likely to have higher BMIs and therefore have a higher cancer risk. In fact, this explanation for the results found by Grismaijer and Singer (1) is proposed in a letter to the editor of The Breast Journal and supported by the findings of an actual peer-reviewed study. This study is oft-cited by people who believe bras increase the risk of breast cancer because they write,
Premenopausal women who do not wear bras had half the risk of breast cancer compared with bra users... (2)but people often neglect to quote the rest of the conclusions:
...possibly because they are thinner and likely to have smaller breasts. Among bra users, larger cup size was associated with an increased risk of breast cancer (P about 0.026), although the association was found only among postmenopausal women and was accounted for, in part, by obesity. These data suggest that bra cup size (and conceivably mammary gland size) may be a risk factor for breast cancer. (2)and
The association, if real, could point to obesity or breast size as the relevant risk factor... in all four study centres bra-wearing controls were substantially heavier than those who did not wear a bra and cup size was strongly correlated with body weight and obesity. (1)
The letter goes on to cite a short experiment they performed. They examined the records of women who had has surgery that is known to reduce lymphatic circulation in the breasts - they found no increased rate of breast cancer in these women. There is no evidence to suggest that reduced lymphatic circulation is correlated with increased cancer risk. In actuality, lymphatic circulation can be the vehicle for metastasis of cancerous cells instead of some magical toxin filter. And we still don't know for sure that bras reduce lymphatic circulation in the first place.
The claim that bras cause breast cancer has also been dismissed by numerous health organizations. There is no need to ask your doctor to find out for sure - the scientific community has swiftly reached a consensus. The idea that bras cause cancer is bullshit pseudoscience scaremongering.
1. T. Gansler and A. Jemal, "Axillary Lymphatic Disruption does not Increase Risk of Breast Carcinoma," Breast Journal, vol. 15 (4), pp 438-439, July/August 2009
2. CC Hsieh and D. Trichopoulos, "Breast size, handedness and breast cancer risk" European Journal of Cancer, vol. 27, pp 131–5. [abstract]