Bubble cavitation is important in technologies such as noninvasive cancer treatment and diagnosis, surface cleaning, and waste-water treatment. The cavitation threshold is the critical external tensile pressure that induces unstable growth of the bubble. Surface nanobubbles have been previously shown experimentally to be stable down to -6 MPa, in disagreement with the Blake threshold, which is the classical cavitation model that predicts bulk bubbles with radii similar to 100 nm should be unstable below -0.6 MPa. Here, we use molecular dynamics to simulate quasi-two-dimensional (2D) and three-dimensional (3D) nitrogen surface nanobubbles immersed in water, subject to a range of pressure drops until unstable growth is observed. We propose and assess new cavitation threshold models, derived from mechanical equilibrium analyses for both the quasi-2D and 3D cavitating bubbles. The discrepancies from the Blake threshold are attributed to the pinned contact line, within which the surface nanobubbles grow with constant lateral contact diameter, and consequently a reduced radius of curvature. We conclude with a critical discussion of previous experimental results on the cavitation of relatively large surface nanobubbles.