Increasing the energy storage capability of lithium-ion batteries necessitates maximization of their areal capacity. This requires thick electrodes performing at near-theoretical specific capacity. However, achievable electrode thicknesses are restricted by mechanical instabilities, with high-thickness performance limited by the attainable electrode conductivity. Here we show that forming a segregated network composite of carbon nanotubes with a range of lithium storage materials (for example, silicon, graphite and metal oxide particles) suppresses mechanical instabilities by toughening the composite, allowing the fabrication of high-performance electrodes with thicknesses of up to 800 μm. Such composite electrodes display conductivities up to 1 × 104 S m−1 and low charge-transfer resistances, allowing fast charge-delivery and enabling near-theoretical specific capacities, even for thick electrodes. The combination of high thickness and specific capacity leads to areal capacities of up to 45 and 30 mAh cm−2 for anodes and cathodes, respectively. Combining optimized composite anodes and cathodes yields full cells with state-of-the-art areal capacities (29 mAh cm−2) and specific/volumetric energies (480 Wh kg−1 and 1,600 Wh l−1).