False-positive test results are among the most common harms of screening tests and may lead to more invasive and expensive diagnostic testing procedures. Estimating the cumulative risk of a false-positive screening test result after repeat screening rounds is, therefore, important for evaluating potential screening regimens. Existing estimators of the cumulative false-positive risk are limited by strong assumptions about censoring mechanisms and parametric assumptions about variation in risk across screening rounds. To address these limitations, we propose a semiparametric censoring bias model for cumulative false-positive risk that allows for dependent censoring without specifying a fixed functional form for variation in risk across screening rounds. Simulation studies demonstrated that the censoring bias model performs similarly to existing models under independent censoring and can largely eliminate bias under dependent censoring. We used the existing and newly proposed models to estimate the cumulative false-positive risk and variation in risk as a function of baseline age and family history of breast cancer after 10 years of annual screening mammography using data from the Breast Cancer Surveillance Consortium. Ignoring potential dependent censoring in this context leads to underestimation of the cumulative risk of false-positive results. Models that provide accurate estimates under dependent censoring are critical for providing appropriate information for evaluating screening tests.