We have discovered an anomalous behavior of CCD readout electronics that affects their use in many astronomical applications. An offset in the digitization of the CCD output voltage that depends on the binary encoding of one pixel is added to pixels that are read out one, two, and/or three pixels later. One result of this effect is the introduction of a differential offset in the background when comparing regions with and without flux from science targets. Conventional data reduction methods do not correct for this offset. We find this effect in 16 of 22 instruments investigated, covering a variety of telescopes and many different front-end electronics systems. The affected instruments include LRIS and DEIMOS on the Keck telescopes, WFC3 UVIS and STIS on HST, MegaCam on CFHT, SNIFS on the UH88 telescope, GMOS on the Gemini telescopes, HSC on Subaru, and FORS on VLT. The amplitude of the introduced offset is up to 4.5 ADU per pixel, and it is not directly proportional to the measured ADU level. We have developed a model that can be used to detect this "binary offset effect" in data, and correct for it. Understanding how data are affected and applying a correction for the effect is essential for precise astronomical measurements.