We report observations with the Chandra X-ray Observatory of the single, cool, magnetic white dwarf GD 356. For consistent comparison with other X-ray observations of single white dwarfs, we also re-analyzed archival ROSAT data for GD 356 (GJ 1205), G99-47(GR 290 = V1201 Ori), GD 90, G195-19 (EG 250 = GJ 339.1), and WD 2316+123, and archival Chandra data for LHS 1038 (GJ 1004) and GD 358 (V777 Her). Our Chandra observation detected no X-rays from GD 356, setting the most restrictive upper limit to the X-ray luminosity from any cool white dwarf: LX < 6.0 × 1025 ergs s-1, at 99.7% confidence, for a 1 keV thermal bremsstrahlung spectrum. The corresponding limit to the electron density is n0 < 4.4 × 10-1 cm-3. Our re-analysis of the archival data confirmed the nondetections reported by the original investigators. We discuss the implications of our and prior observations for models of coronal emission from white dwarfs. For magnetic white dwarfs, we emphasize the more stringent constraints imposed by cyclotron radiation. In addition, we describe (in an Appendix) a statistical methodology for detecting a source and for constraining the strength of a source, which applies even when the number of source or background events is small. © 2007. The American Astronomical Society. All rights reserved.