That would only help if the false positive occur completely at random. If the error is systematic (for example because the test is detecting something else in the blood) you'd just keep getting false positives.

presumably these other things that would cause repeated false positives would be identified over time and could be ruled out with other tests - ie you either have cancer or a serious iron deficiency, let's check if you have an iron deficiency.

It probably depends on what is causing the false positive.

If it is just some variance in the test, then multiple testings could help (And the test could probably be improved to reduce false positives pretty easily)

But if there is something intrinsic to those tested (say, some hormone level or other) Repeat tests will give repeat false positives. AND it may be harder to eliminate false positives by refining the test