Static structure factors (S(q)) for many ionic liquids show low-wavenumber peaks whose intensities increase with increasing temperature. The greater peak intensities might seem to imply increasing intermediate-range order with increasing temperature. Molecular dynamics (MD) simulations for a representative ionic liquid, 1-butyl-1-methylpyrrolidinium bis(trifluoromethylsulfonyl)imide (C(4)C(1)pyrrTFSI), were used to calculate S(q) and partial S(q) (cation-cation, anion-anion, and cation-anion) at 298, 363, and 500 K. S(q) and partial S(q) were further decomposed into positive and negative components (which each indicate structural ordering) by separately summing positive and negative Fourier transform summands. Increasing temperature causes the negative components of each partial S(q) to decrease in magnitude more than the positive components, causing the total S(q) to increase in magnitude. Thus, structural ordering with periodicities corresponding to observed peaks in S(q) does not increase but instead decoheres with increasing temperature, even though S(q) peak heights increase. Fourier transform summands also show where in real space the positive and negative component contributions to S(q) change when the temperature increases. This new, detailed analysis based on Fourier transform summands comprising S(q) argues for great caution when interpreting S(q) intensities and highlights the value of simulations as a complement to X-ray (or neutron) scattering experiments.