The amount of light produced by nuclear recoils in scintillating targets is strongly quenched compared to that produced by electrons. A precise understanding of the quenching factor is particularly interesting for weakly interacting massive particles (WIMP) searches and coherent elastic neutrino-nucleus scattering (CE⁢𝜈⁢NS) measurements since both rely on nuclear recoils, whereas energy calibrations are more readily accessible from electron recoils. There is a wide variation among the current measurements of the quenching factor in sodium iodide (NaI) crystals, especially below 10 keV, the energy region of interest for dark matter and CE⁢𝜈⁢NS studies. A better understanding of the quenching factor in NaI(Tl) is of particular interest for resolving the decades-old puzzle in the field of dark matter between the null results of most WIMP searches and the claim for dark matter detection by the DAMA/LIBRA collaboration. In this work, we measured sodium and iodine quenching factors for five small NaI(Tl) crystals grown with similar thallium concentrations and growth procedures. Unlike previous experiments, multiple crystals were tested, with measurements made in the same experimental setup to control systematic effects. The quenching factors agree in all crystals we investigated, and both sodium and iodine quenching factors are smaller than those reported by DAMA/LIBRA. The dominant systematic effect was due to the electron equivalent energy calibration originating from the nonproportional behavior of the NaI(Tl) light yield at lower energies, potentially the cause for the discrepancies among the previous measurements.