Today's technical systems are complex far beyond their designers' intentions because of the vast array of interactions, both intended and unintended, with other systems, with their environments, and (perhaps most complex of all) with us humans. The term Interwoven Systems has been developed to describe these entities. Proposals to integrate and manage such large, intensively-interconnected, and constantly evolving infrastructures involve adding both computational tasks and stored data. This dynamic ecosystem, exemplified by the Internet of Things (IoT), contains not only major resources such as email systems and the electrical power grid, but also internet-connected commercial appliances, routers, and other devices that are not considered to be computational systems by their owners. Both minimal hardware and excess capacity pose potential risks for the larger interwoven system in which they reside, most especially because their human users and operators frequently do not understand the full complexity of results which their interactions with these resources can have. This position paper discusses these systemic risks and proposes possible approaches for identifying and managing them.