UNAIDS established fast-track targets of 73% and 86% viral suppression among human immunodeficiency virus (HIV)-positive individuals by 2020 and 2030, respectively. The epidemiologic impact of achieving these goals is unknown. The HIV-Calibrated Dynamic Model, a calibrated agent-based model of HIV transmission, is used to examine scenarios of incremental improvements to the testing and antiretroviral therapy (ART) continuum in South Africa in 2015. The speed of intervention availability is explored, comparing policies for their predicted effects on incidence, prevalence and achievement of fast-track targets in 2020 and 2030. Moderate (30%) improvements in the continuum will not achieve 2020 or 2030 targets and have modest impacts on incidence and prevalence. Improving the continuum by 80% and increasing availability reduces incidence from 2.54 to 0.80 per 100 person-years (-1.73, interquartile range (IQR): -1.42, -2.13) and prevalence from 26.0 to 24.6% (-1.4 percentage points, IQR: -0.88, -1.92) from 2015 to 2030 and achieves fast track targets in 2020 and 2030. Achieving 90-90-90 in South Africa is possible with large improvements to the testing and treatment continuum. The epidemiologic impact of these improvements depends on the balance between survival and transmission benefits of ART with the potential for incidence to remain high.