Most cellular systems, from macromolecules to genetic networks, have more than one function. Examples involving networks include the transcriptional regulation circuits formed by Hox genes and the Drosophila segmentation genes, which function in both early and later developmental events. Does the need to carry out more than one function severely constrain network architecture? Does it imply robustness trade-offs among functions? That is, if one function is highly robust to mutations, are other functions highly sensitive, and vice versa? Little available evidence speaks to these questions. We address them with a general model of transcriptional regulation networks. We show that requiring a regulatory network to carry out additional functions constrains the number of permissible network architectures exponentially. However, robustness of one function to regulatory mutations is uncorrelated or weakly positively correlated to robustness of other functions. This means that robustness trade-offs generally do not arise in the systems we study. As long as there are many alternative network structures, each of which can fulfill all required functions, multiple functions may acquire high robustness through gradual Darwinian evolution.