As is true of many questions and answers, it depends...
In the case of cancer screening (mammogram, colonoscopy, etc.) and many other screening tests for a disease or condition, this is almost always the case. For a screening test to have some value, it must be "sensitive" enough to detect the relatively rare cases (say 1% or sometimes much less) of the condition being screened. The true positive fraction (TPF) is almost always less than the false positive fraction (FPF).
That is why there is always a retest (applying the same test again) or follow up tests (likely more expensive but higher "specificity"), to then eliminate the false positives.
So in a sense the name you are asking for is "screening test"!
The term "accuracy" has a very particular technical meaning, which is not necessarily the common meaning, or commonly thought of situation. Most "common sense" is related to a 50% 50% chance, you have cancer or you don't.
From the wiki page: https://en.wikipedia.org/wiki/Receiver_operating_characteristic
Another way of putting it is that a test is accurate if it gets most cases correct. Which is the common definition. But if the condition is rare, and the test is "sensitive" it can (and in fact should and must) still give false positives.
1% prevalence, 1000 tests, 10 true positives, 20 false positives
accuracy = (10 + (1000 - 10 - 20))/1000 = 98%
Yet another technical way of saying this is that screening tests tend to operate at the high sensitivity (high false positive) side of the so called receiver operating characteristic (ROC). One wants to catch all the true positives, at the expense of false positives, which will be retested and largely eliminated.