Complementary metal–oxide–semiconductor (CMOS) logic circuits at their ultimate scaling limits place extreme demands on the properties of all materials involved. The requirements for semiconductors are well explored and could possibly be satisfied by a number of layered two-dimensional (2D) materials, such as transition metal dichalcogenides or black phosphorus. The requirements for gate insulators are arguably even more challenging. At present, hexagonal boron nitride (hBN) is the most common 2D insulator and is widely considered to be the most promising gate insulator in 2D material-based transistors. Here we assess the material parameters and performance limits of hBN. We compare experimental and theoretical tunnel currents through ultrathin layers (equivalent oxide thickness of less than 1 nm) of hBN and other 2D gate insulators, including the ideal case of defect-free hBN. Though its properties make hBN a candidate for many applications in 2D nanoelectronics, excessive leakage currents lead us to conclude that hBN is unlikely to be suitable for use as a gate insulator in ultrascaled CMOS devices.