We present a theoretical study of the intrinsic quality factor of the fundamental flexural vibration in a carbon nanotube and its dependence on temperature, radius, length, and tension. In particular, we examine three- and four-phonon decays of the fundamental flexural mode within quantized elasticity theory. This analysis reveals design principles for the construction of ultrasensitive nanotube mass sensors: under tensions close to the elastic limit, intrinsic losses allow for single yoctogram mass resolution at room temperature while cooling opens the possibility of subyoctogram mass resolution. © 2010 The American Physical Society.