It appears that there is confusion between different measures here.
First, I am focussed on the PCR tests (i.e. trying to detect fragments of the virus) rather than the serology test (i.e. trying to detect the body's reaction to the virus.)
Now, how do we measure the accuracy?
One measure is the "specificity".
Specificity (True Negative rate) measures the proportion of negatives that are correctly identified (i.e. the proportion of those who do not have the condition (unaffected) who are correctly identified as not having the condition).
The specificity of the PCR test is going to depend on the lab and details of the test (e.g. the cycle-threshold, Ct), but it can generally be considered high.
RT-PCR assays in the UK have analytical sensitivity and specificity of greater than 95%
An unpeer-reviewed pre-print, Diagnosing SARS-CoV-2 infection: the danger of over-reliance on positive
test results from September 2020 warns that the real-world figures may be lower than than 100% (that they claim others claim), and go on to express concern about how the authorities have interpreted these results. The figures from the studies they do cite include 97-99.7%, >93.7%, >99.6%, 99.3%.
So, even a paper concerned that the specificity is too low showed very high numbers.
So, to summarise: If a random uninfected person takes a PCR test to try to detect if there are parts of the virus in their body, the chance that they will told that the test wrongly said "Yes" is pretty low.
Another measure is the "Positive Predictive Value."
The positive predictive values (PPV) is the proportion of positive results in statistics and diagnostic tests that are true positive results. The PPV describes the performance of a diagnostic test or other statistical measure. A high result can be interpreted as indicating the accuracy of such a statistic. The PPV is not intrinsic to the test (as true positive rate and true negative rate are); it depends also on the prevalence.
[This isn't a direct-quote. I removed an irrelevant definition pf NPV that was woven in.]
So this is the figure that shows how worried you should be if you receive a positive result.
This is an area that is hard for the public (and even doctors) to get their heads around. Gerd Gigerenzer's book "Reckoning with Risk" warns about widespread innumeracy in this area, and proposes techniques (e.g. frequency trees) to help resolve it.
If I can give an example using Gigerenzer's technique: I live in a state in Australia where we have been very fortunate recently to have a very low prevalence of COVID-19 and generally limited to travellers in quarantine and their immediate contacts. Let's say, for illustration, that the prevalence is around 3 per million. Let's assume the sensitivity was 100% and specificity was at the high end of the estimates given: 99.6%.
That means if 1 million people were tested, 3 would get a true positive test because they were infected, and 4000 would get a false positive test (0.4% of 1,000,000). Only 0.075% of positive tests would be true positives!
For as long as the virus does not escape its containment (again!) and we don't have another outbreak, as an Australian, who is not a traveller and not in close contact with any, both of these statements are true, and not in conflict with each other:
- I can be confident that if I am uninfected and I get tested, I will not get a false positive - thank you, high specificity.
- I can be confident that if I do get a positive, it is very likely a false positive - thank you, low prevalence leading to a low positive predictive value.
These figures would be very different if I was from an area where the prevalence was much higher. I don't offer these calculations as accurate and correct, but as an illustration of how even a low false-positive rate can overwhelm a test in an area of low-prevalence.
What is the claim?
The original claim in German sources the Wadsworth Center's report. Which definition was it using?
I was unable to find the original, but the clearest explanation I found of the contents was from a testimonial referencing a (now missing) newspaper report.
At Mandavilli’s request, New York state’s Wadsworth Center examined 872 positive PCR test results it had obtained in July, after amplification for 40 cycles. “With a cutoff of 35,”
Mandavilli reported, “about 43 percent of those tests would no longer qualify as positive. About 63 percent would no longer be judged as positive if the cycles were limited to 30.”
It is clear that they are not referring to specificity, but to Positive Predictive Value.
In summary: There is no conflict between the two statements quoted in the question.