The nonadditive kinetic potential 𝑣^NAD is a key quantity in density-functional theory (DFT) embedding methods, such as frozen density embedding theory and partition DFT. 𝑣^NAD is a bifunctional of electron densities 𝜌B and 𝜌tot=𝜌A+𝜌B. It can be evaluated using approximate kinetic-energy functionals, but accurate approximations are challenging. The behavior of 𝑣^NAD in the vicinity of the nuclei has long been questioned, and singularities were seen in some approximate calculations. In this article, the existence of singularities in 𝑣^NAD is analyzed analytically for various choices of 𝜌B and 𝜌tot, using the nuclear cusp conditions for the density and Kohn-Sham potential. It is shown that no singularities arise from smoothly partitioned ground-state Kohn-Sham densities. We confirm this result by numerical calculations on diatomic test systems HeHe, HeLi⁺, and H₂, using analytical inversion to obtain a numerically exact 𝑣^NAD for the local density approximation. We examine features of 𝑣^NAD which can be used for development and testing of approximations to 𝑣^NAD⁡[𝜌B,𝜌tot] and kinetic-energy functionals.