Clathrin is a naturally evolved protein that robustly assembles and disassembles into nanoscale spherical cages. This ability to reorganize in a highly dynamic fashion makes clathrin an attractive model system to study the kinetic and thermodynamic principles of biomolecular self-assembly. Through a combination of experimental and computational approaches, we demonstrate that competition between weak non-specific and specific reversible interactions can dictate the initial pathway of the assembly process, yet the final assembled structures are not sensitive to this competition. We conclude that the relative strengths of non-specific and specific interactions control clathrin assembly at short time scales resulting in either disordered protein aggregates or regularly structured assemblies. However with sufficient time for remodeling, the final assembled structure is robustly formed due to geometric constraints arising from specific molecular recognition events. These data provide insight into naturally evolved biological assembly processes and guidance for the design of engineered systems to achieve robust assembly.