Synthetic biology seeks to create complex systems in living organisms modularly. Unfortunately, modularity is hindered by several factors. One major factor limiting the scalability of rationally engineered large-scale genetic circuits is unwanted coupling among modules due to competition for shared cellular resources. Leveraging a mechanistic model explicitly accounting for the limited availability of these resources, in this paper we reveal how competition for shared resources affects the stability profile of the toggle switch, one of the most widespread genetic modules. As a result, we uncover the connection between parameter asymmetry, resource sequestration and bistability, not only in the case of a single toggle switch, but also when multiple modules all share the same pool of resources. To demonstrate the relevance of our results, we illustrate when and why the collective behavior of bistable toggle switches becomes monostable, and reveal how the interplay between parameter asymmetry and resource sequestration contributes to the emergence of this surprising phenomenon.