How common is breast implant illness?

Thank you for your question. I can only answer your question by saying that not only is it "not common," but I would also caution you that "breast implant illness" does not exist. This is a anecdotal (case by case report by individual patients) phenomenon that has no scientific basis and has been researched in depth and shown to have no connection to breast implant surgery. In other words, a patient gets implants, and then gets diagnosed with an illness of some kind which they attribute to the implants, so they go around telling everyone they have "breast implant illness." Unfortunately, there is no way to know whether the patient would have developed that illness with or without the implants. If you look at the general population, the incidence of those illnesses people attribute to implants is roughly equivalent to the rate of that same illness in patients who do not have implants. Thus the scientific conclusion is that regardless of what that person thinks ("I got breast implant illness"), there is no scientifically proven evidence that the implant is associated in any way with the development of the illness. Bottom line is stop reading all the non-sense you are reading and stop listening to people who are pushing this agenda and start looking at the scientific data which takes into account thousands of patients over several years of time. This should give you confidence to proceed with surgery and feel safe with that decision. Hope this helps!

Silicone implants are safe. Period
Breast Implant Illness  and silicone related medical complications are controversial topics. It is a phenomenon which is supported by a wealth of anecdotal data (patient testimony) but no empiric data  (hard scientific research). Patient reports a litany of symptoms which they attribute to their implants but the connection is difficult if not impossible to prove via laboratory study.
Physician scientists may recognize anecdotal data but unless it is supported by Class I empiric data, it is difficult to place a great deal of faith in it. Clinicians make decision based upon evidence based medicine. 
That being said, it is easier to identify what one knows rather than what one doesn't.
What we do know is that by all available measures today...implants are safe and specifically silicone implants are safe.
Silicone breast implants are among the safest medical devices in use today. These devices are also among the most rigorously studied and tested medical devices currently being used in the U.S. today. It is biologically inert  and in that respect no different than the titanium used for bony fixation. There has been no demonstrated link between silicone and the commonly cited myth of autoimmune disease.
With regards to your specific question, ultimately it is up to patients to decide what sources they choose to believe. Those who believe they suffer from BII are vocal and the accounts of their suffering are readily available to most (via the internet). The sterile scientific data regarding silicone implant safety is by contrast less available and not as easily consumed by the average individual. That being said the hard scientific data is no less voluminous. There are as many (if not more) fact based studies and data points as there are those who believe they are suffering from BII. 
As always, discuss your concerns with a board certified plastic surgeon (ABPS)