A series of deep stratospheric intrusions in late May 2013 increased the daily maximum 8 h surface ozone (O3) concentrations to more than 70 parts per billion by volume (ppbv) at rural and urban surface monitors in California and Nevada. This influx of ozone‐rich lower stratospheric air and entrained Asian pollution persisted for more than 5 days and contributed to exceedances of the 2008 8 h national ambient air quality standard of 75 ppbv on 21 and 25 May in Clark County, NV. Exceedances would also have occurred on 22 and 23 May had the new standard of 70 ppbv been in effect. In this paper, we examine this episode using lidar measurements from a high‐elevation site on Angel Peak, NV, and surface measurements from NOAA, the Clark County, Nevada Department of Air Quality, the Environmental Protection Agency Air Quality System, and the Nevada Rural Ozone Initiative. These measurements, together with analyses from the National Centers for Environmental Prediction/North American Regional Reanalysis; NOAA Geophysical Fluid Dynamics Laboratory AM3 model; NOAA National Environmental Satellite, Data, and Information Service Real‐time Air Quality Modeling System; and FLEXPART models, show that the exceedances followed entrainment of ~20 to 40 ppbv of lower stratospheric ozone mingled with another 0 to 10 ppbv of ozone transported from Asia by the unusually deep convective boundary layers above the Mojave desert. Our analysis suggests that this vigorous mixing can affect both high and low elevations and help explain the springtime ozone maximum in the southwestern U.S. Plain Language Summary Ozone (O3) is formed high in the upper atmosphere when ultraviolet (UV) rays from the sun break the bond in molecular oxygen (O2). This so‐called "good" ozone protects plants and animals at the ground from the damaging effects of these UV rays. Ozone is also formed in the lower atmosphere through reactions between nitrogen oxides (NOx) and volatile organic compounds (VOCs) of natural and human origin. This "bad" ozone can damage the sensitive tissues of plants and animals. The U.S. Clean Air Act (CAA) established standards designed to keep the ozone in the air we breathe at healthy levels by regulating NOx and VOC precursors emitted by human activities. The standard is revised periodically as new data on the health effects of ozone become available, and the current standard is close to the normal background levels found in some parts of the U.S. during springtime. This means that the concentrations can sometimes exceed the standard when storms bring "good" ozone from the upper atmosphere down to the ground where it becomes "bad" ozone. We take a look at this process and discuss why it is more likely to happen in the Southwest than in other parts of the U.S. Key Points The deep convective boundary layers of the Southwestern U.S. increase entrainment of stratospheric air and transported pollution Stratospheric intrusions contribute to the springtime surface ozone maximum in the Southwestern U.S. Stratospheric intrusions can cause exceedances of the National Ambient Air Quality Standard for ozone in the Southwestern U.S.