The short answer is "yes".
Here is another description of (I think) the same studies from Leonard Mlodinow's "The Drunkard's Walk":
[...] in studies in Germany and the United States, researchers asked physicians to estimate the probability that an asymptomatic woman between the ages of 40 and 50 who has a positive mammogram actually has breast cancer if 7 percent of mammograms show cancer when there is none. In addition, the doctors were told that the actual incidence was about 0.8 percent and that the false-negative rate about 10 percent. Putting that all together, one can use Bayes’s methods to determine that a positive mammogram is due to cancer in only about 9 percent of the cases. In the German group, however, one-third of the physicians concluded that the probability was about 90 percent, and the median estimate was 70 percent. In the American group, 95 out of 100 physicians estimated the probability to be around 75 percent.
Similar results were reported by (smaller) studies from 1978 to 2014.
The problem in these cases is conditional probability. Generally, conditional probability is the probability of one event given another event (or condition). For example, the probability of throwing two sixes with two six-sided dice is 1 in 36; however, the conditional probability of throwing two sixes with two six-sided dice if the first one already turned out to show a six is just 1 in 6.
With any medical test, you have questions of conditional probability. Often, the number reported about a test is its accuracy; for example, an HIV test may produce the correct result 99% of the time. This means that if you have HIV, you have a 99% chance that you will be correctly diagnosed as HIV positive; and if you do not, you have a 99% chance of being correctly diagnosed as HIV negative. The fallacy is to assume that this means that if you are diagnosed with HIV, there is a 99% chance that you are indeed HIV positive. In reality, the probability is much lower and depends heavily on your risk group; it might well be 10%, or 1%, or even below that (I do not want to go into the calculation details here).
The mistake is to assume that the probability of you being correctly diagnosed when positive and the probability of you being positive when diagnosed are the same. You can see this on an obvious example: If you are a professional football player, you are extremely likely to be a male. However, even if you are a male, you are still extremely unlikely to be a professional football player. The reason for this is that there are quite a lot of men, and very few football players; the same applies for the HIV example, since very few people are HIV positive in the first place.
Does this happen in practice?
Finally - a chance to use anecdotal evidence with (relative) impunity! Yes, it does happen. Mlodinow, in his aforementioned book, writes about a similar case where he received a false HIV diagnosis. And I myself had a family member diagnosed with a "definitely malignant" tumor. And - as a side note - although I read about the issue shortly before, it did not occur to me to doubt the doctor's judgement until after the operation showed that the tumor was benign.
Is this about doctors?
No. Doctors are subject to this fallacy, and of course they are a group that is not supposed to err. But people generally have bad intuitions of many probabilistic tasks, this one included. Indeed, the common name for this particular fallacy is the prosecutor's fallacy, which gives a hint as to who else might be affected. The linked Wikipedia article has explanations and examples of legal cases where this played a crucial (and very destructive) role.
Can't you trust doctors now?
Well, can you trust anyone? I expect most doctors to be competent at their core tasks. However, they are just people, and most of them have not been explicitly trained in probability theory. Therefore, you should use healthy skepticism when confronted with numbers describing risks and probability calculations, whether they come from doctors, lawyers and others, sometimes even mathematicians.