The potential role of bottom-up Synthetic Cells (SCs) in the Internet of Bio-Nano Things (IoBNT) is discussed. In particular, this perspective paper focuses on the growing interest in networks of biological and/or artificial objects at the micro- and nanoscale (cells and subcellular parts, microelectrodes, microvessels, etc.), whereby communication takes place in an unconventional manner, i.e., via chemical signaling. The resulting "molecular communication" (MC) scenario paves the way to the development of innovative technologies that have the potential to impact biotechnology, nanomedicine, and related fields. The scenario that relies on the interconnection of natural and artificial entities is briefly introduced, highlighting how Synthetic Biology (SB) plays a central role. SB allows the construction of various types of SCs that can be designed, tailored, and programmed according to specific predefined requirements. In particular, "bottom-up" SCs are briefly described by commenting on the principles of their design and fabrication and their features (in particular, the capacity to exchange chemicals with other SCs or with natural biological cells). Although bottom-up SCs still have low complexity and thus basic functionalities, here, we introduce their potential role in the IoBNT. This perspective paper aims to stimulate interest in and discussion on the presented topics. The article also includes commentaries on MC, semantic information, minimal cognition, wetware neuromorphic engineering, and chemical social robotics, with the specific potential they can bring to the IoBNT.