Severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) antibody tests of varying specificity and sensitivity are now available. For informing individuals whether they have had coronavirus disease 2019 (COVID-19), they need to be very accurate. For measuring population prevalence of past infection, the numbers of false positives and negatives need to be roughly equal. With a series of worked examples for a notional population of 100,000 people, we show that even test systems with a high specificity can yield a large number of false positive results, especially where the population prevalence is low. For example, at a true population prevalence of 5%, using a test with 99% sensitivity and specificity, 16% of positive results will be false and thus 950 people will be incorrectly informed they have had the infection. Further confirmatory testing may be needed. Giving false reassurance on which personal or societal decisions might be based could be harmful for individuals, undermine public confidence and foster further outbreaks.