We report measurements of field-effect transistors made of isolated single-walled carbon nanotubes contacted by superconducting electrodes. For large negative gate voltage, we find a dip in the low-bias differential resistance. Remarkably, this dip persists well above the superconducting transition temperature of the electrodes, indicating that it is not caused by superconducting proximity effect from the electrodes. This conclusion is supported by measurements on carbon nanotubes contacted by normal electrodes showing similar features. One possible explanation is superconductivity in the nanotubes, occurring when the gate voltage shifts the Fermi energy into van Hove singularities of the electronic density of states.