This paper investigates the dependence of breakdown pressure, the critical pressure at which tensile failure of the rock is initiated by injecting fluid in a borehole, on the rate of pressurization. The mathematical model explicitly accounts for the existence of micro-cracks at the borehole wall that trigger the failure process. Breakdown, in this context, occurs when the stress intensity factor of a critically oriented micro-crack reaches the rock toughness. The model is presently restricted to low-permeability/low-porosity rocks. By considering one-dimensional lubrication flow in the crack coupled with the non-local elastic response of the crack, the evolution of the net pressure, crack opening and stress intensity factor is obtained as functions of the pressurization rate. The relation between breakdown pressure and pressurization rate in the case of zero initial net pressure is shown to be controlled by only one dimensionless number: the ratio between the initial width of the unstressed micro-crack and the induced elastic opening at failure. It is further shown that (i) the fluid pressure in the early stages of the pressurization history drops in the crack and that cavitation can occur, and (ii) local back-flow in the crack takes place. The dependence of breakdown pressure, pb, on the pressurization rate, A, is determined as well as the range of A, where pb varies significantly. The lower and pseudo upper bounds of this range of pressurization rate correspond to limiting regimes of slow and pseudo fast pressurization.