Basically, an extremely low rate of incidence (in Cory's example, 1 in a million) will render a "highly accurate" test meaningless.
That's a huge increase from a 0.0001% chance of having the disease, but it's still not flat out terrifying. Repeat testing can weed through the false positives at a speed proportional to its accuracy.
If the test is picking up something in the person being tested, then yes, you'll get the same result every time and repeated testing proves nothing. But you can still repeat using other tests.
If the test gives false positives purely at random, then repeated testing will help. Say the test is wrong 50% of the time, and you do the test five times. If you get the same results every time, then you can be 100-(50/100)^5*100 = 97% sure of the results.
No, it means that the "99% accurate" test is wrong 9,999 times out of 1,000,000. It would be clear to anyone when stated that way. What's counterintuitive is the author's statement of the result, not the result itself.