"Accurate" is too broad a term. The hospital test was likely very accurate, it was just optimized for sensitivity (being sure to catch it if it's there, at the expense of false positives), rather than specificity (being able to have a high degree of confidence that a positive test is true, at the expense of a higher false negative rate).

It is _very_ common in medicine to use a very sensitive test to screen for conditions, and then cross check any positives with a more specific test (which is what should have been done in this case, prior to taking any action).