The electrical noise characteristics of single-walled carbon nanotubes have been investigated. For all three cases of individual isolated nanotubes, thin films of interconnected nanotubes, and bulk nanotube mats, anomalously large bias-dependent 1/f noise is found. The noise magnitude greatly exceeds that commonly observed in metal films, carbon resistors, or even carbon fibers with comparable resistances. A single empirical expression describes the noise for all nanotube samples, suggesting a common noise-generating mechanism proportional only to the number of nanotubes in the conductor. We consider likely sources of the fluctuations, and consequences for electronic applications of nanotuhes if the excessive noise cannot be suppressed.