Nanopore fabrication by controlled breakdown (CBD) overcomes many of the challenges of traditional nanofabrication techniques, by reliably forming solid-state nanopores sub-2 nm in size in a low-cost and scalable way for nucleic acid analysis applications. Herein, the breakdown kinetics of thin dielectric membranes immersed in a liquid environment are investigated in order to gain deeper insights into the mechanism of solid-state nanopore formation by high electric fields. For various fabrication conditions, we demonstrate that nanopore fabrication time is Weibull-distributed, in support of the hypothesis that the fabrication mechanism is a stochastic process governed by the probability of forming a connected path across the membrane (i.e. a weakest-link problem). Additionally, we explore the roles that various ions and solvents play in breakdown kinetics, revealing that asymmetric pH conditions across the membrane can significantly affect nanopore fabrication time for a given voltage polarity. These results, characterizing the stochasticity of the nanopore fabrication process and highlighting the parameters affecting it, should assist researchers interested in exploiting the potential of CBD for nanofluidic channel fabrication, while also offering guidance towards the conceivable manufacturing of solid-state nanopore-based technologies for DNA sequencing applications.