JWST transmission and emission spectra will provide invaluable glimpses of transiting exoplanet atmospheres, including possible biosignatures if photon-limited performance can be achieved. This promising science from JWST, however, will require exquisite precision and understanding of systematic errors that can impact the time series of planets crossing in front of and behind their host stars. We find that 1/f noise can limit the precision of grism time series, but will average down like the square root of N frames/reads. The current NIRCam instrument’s grism time series mode is especially affected by 1/f noise because its GRISMR dispersion direction is parallel to the detector fast-read direction, but could be alleviated in the GRISMC direction. For bright sources observed with GRISMR, the 1/f noise can potentially be larger than photon noise. By contrast, we find that systematic errors are very manageable. In total, our estimated noise floor from known systematic error sources is only 9 ppm per visit. The dominant contributor is pointing jitter and high gain antenna moves on top of the detectors’ subpixel crosshatch patterns that will produce relatively small variations (less than 6 parts per million, ppm). To achieve these low noise sources, it is important to employ a sufficiently large (more than 1.1 arcseconds) extraction aperture. We do however urge caution as unknown systematic error sources could be present in flight and will only be measurable on astrophysical sources like quiescent stars. Finally, we note that there may be some perennial instrument offsets at the 40 ppm level due to detector non-linearities and reciprocity failure, so corrections may be needed when stitching together a multi-instrument multi-observatory spectrum over wide wavelength ranges.