Electropermeabilization or electroporation is the electrical disruption of a cell's membrane to introduce drugs, DNA/RNA, proteins, or other therapies into the cell. Despite four decades of study, the fundamental science of the process remains poorly understood and controversial. We measured the minimum applied electric field required for permeabilization of suspended spherical cells as a function of the cell radius for three cell lines. Key to this work is our use of optical tweezers to precisely position individual cells and enable well-defined, repeatable measurements on cells in suspension. Our findings call into question fundamental assumptions common to all theoretical treatments that we know of. It is generally expected that, for individual cells from a particular cell line, large cells should be easier to electroporate than small ones: the minimum electric field to cause electropermeabilization should scale inversely with the cell diameter. We found instead that each cell line has its own characteristic field that will, on average, cause permeabilization in cells of that line. Electropermeabilization is a stochastic process: two cells which appear identical may have different permeabilization thresholds. However, for all three cell lines, we found that the minimum permeabilization field for any given cell does not depend on its size.