We explore the magnetization reversal process of individual Co nanoislands grown on Cu(111) by low temperature spin-polarized scanning tunneling microscopy (spin-STM) and spectroscopy (spin-STS). We measure hysteresis loops of the differential conductance of single Co islands in magnetic fields of up to 4 T. From such hysteresis loops we extract the magnetic switching field of single Co islands as a function of island size. Tentatively we analyze the size dependence of the switching field using the venerable model of thermally assisted coherent magnetization reversal. We present evidence for the failure of that model to explain our experimental results. We propose that the magnetization reversal process within individual Co nanoislands on Cu(111) is a non-coherent process.