Adaptive behavior requires the ability to focus on a current task and protect it from distraction (cognitive stability), as well as the ability to rapidly switch to another task in light of changing circumstances (cognitive flexibility). Cognitive stability and flexibility have been conceptualized as opposite endpoints on a stability-flexibility trade-off continuum, implying an obligatory reciprocity between the two: Greater flexibility necessitates less stability, and vice versa. Surprisingly, rigorous empirical tests of this critical assumption are lacking. Here, we acquired simultaneous measurements of cognitive stability (congruency effects) and flexibility (switch costs) on the same stimuli within the same task while independently varying contextual demands on these functions with block-wise manipulations of the proportion of incongruent trials and task switches, respectively. If cognitive stability and flexibility are reciprocal, increases in flexibility in response to higher switch rates should lead to commensurate decreases in stability, and increases in stability in response to more frequent incongruent trials should result in decreased flexibility. Across three experiments, using classic cued task-switching (Experiments 1 and 3) and attentional set-shifting (Experiment 2) protocols, we found robust evidence against an obligatory stability-flexibility trade-off. Although we observed the expected contextual adaptation of stability and flexibility to changing demands, strategic adjustments in stability had little influence on flexibility, and vice versa. These results refute the long-held assumption of a stability-flexibility trade-off, documenting instead that the cognitive processes mediating these functions can be regulated independently-it is possible to be both stable and flexible at the same time.