Many modern products exhibit high reliability under normal operating conditions. Conducting life tests under these conditions may result in very few observed failures, insufficient for accurate inferences. Instead, accelerated life tests (ALTs) must be performed. One of the most popular ALT designs is the step-stress test, which shortens the product's lifetime by progressively increasing the stress level at which units are subjected to at some pre-specified times. Classical estimation methods based on the maximum likelihood estimator (MLE) enjoy suitable asymptotic properties but they lack robustness. That is, data contaminationcan significantly impact the statistical analysis. In this paper, we develop robust inferential methods for highly reliable devices based on the density power divergence (DPD) for estimating and testing under the step-stress model with intermittent monitoring and Weibull lifetime distributions. We theoretically and empirically examine asymptotic and robustness properties of the minimum DPD estimators and associated Wald-type test statistics. Moreover, we develop robust estimators and confidence intervals for some important lifetime characteristics. The effect of temperature in solar lights, medium power silicon bipolar transistors and LED lights using real data arising from an step-stress ALT is analyzed applying the robust methods proposed.