Increasing electric vehicle (EV) charging demand and residential solar photovoltaic (PV) generation greatly alter traditional distribution system operation and have the potential to overload and otherwise threaten the operating life of legacy infrastructure. The rate and location of adoption of these technologies on residential distribution systems introduce operational uncertainties for which traditional utilities may not be prepared. This paper proposes a user-defined, data-driven risk assessment method to quantify the severity and likelihood of transformer and secondary conductor overload conditions posed by high levels of EV charging demand coupled with rooftop solar PV. The stochasticity inherent in the operation of a distribution secondary system is captured by employing data-driven probability distribution functions for residential loading, EV charging, rooftop solar generation and ambient temperature. Samples then are repeatedly drawn from each function as inputs to a Monte-Carlo, multi-period power flow analysis to calculate secondary line currents, total loading and accelerated transformer aging. The proposed approach is utilized to study transformer and secondary conductor overload risk as well as transformer loss-of-life for multiple EV and PV penetration scenarios using historical EV charging profiles and residential Salt Lake City load profile data for a peak summer load day.