I will present theoretical support for a hypothesis about cell-cell contact, which plays a critical role in immune function. A fundamental question for all cell-cell interfaces is how receptors and ligands come into contact, despite being separated by large molecules, the extracellular fluid, and other structures in the glycocalyx. The cell membrane is a crowded domain filled with large glycoproteins that impair interactions between smaller pairs of molecules, such as the T cell receptor and its ligand, which is a key step in immunological information processing and decision-making. A first passage time problem allows us to gauge whether a reaction zone can be cleared of large molecules through passive diffusion on biologically relevant timescales. I combine numerical and asymptotic approaches to obtain a complete picture of the first passage time, which shows that passive diffusion alone would take far too long to account for experimentally observed cell-cell contact formation times. The result suggests that cell-cell contact formation may involve previously unknown active mechanical processes.