A test can catch every case and still produce mostly false alarms when the condition is rare.
Suppose 1 in 1,000 people has a condition. A test catches every case, but also flags 5% of healthy people. In a group of 1,000, that is about one true positive and 50 false positives. So among roughly 51 positive results, only one person has the condition: about a 2% chance.
The calculation does not say the test is useless. It shows why accuracy is not one number: sensitivity, false-positive rate, and prevalence all matter. The result changes when prevalence or the test’s performance changes. In real medicine, a clinician interprets the test alongside symptoms, history, and sometimes a confirmatory test.