Lattice-based constructs, often made by additive manufacturing, are attractive for many applications. Typically, such constructs are made from microscale or larger elements; however, smaller nanoscale components can lead to more unusual properties, including greater strength, lighter weight, and unprecedented resiliencies. Here, solid and hollow nanoparticles (nanoframes and nanocages; frame size: ~15 nanometers) were assembled into colloidal crystals using DNA, and their mechanical strengths were studied. Nanosolid, nanocage, and nanoframe lattices with identical crystal symmetries exhibit markedly different specific stiffnesses and strengths. Unexpectedly, the nanoframe lattice is approximately six times stronger than the nanosolid lattice. Nanomechanical experiments, electron microscopy, and finite element analysis show that this property results from the buckling, densification, and size-dependent strain hardening of nanoframe lattices. Last, these unusual open architectures show that lattices with structural elements as small as 15 nanometers can retain a high degree of strength, and as such, they represent target components for making and exploring a variety of miniaturized devices.