One illustrative method disclosed herein includes, among other things, forming a plurality of trenches in a semiconductor substrate to thereby define a plurality of fins in the substrate, forming a layer of insulating material in the trenches, performing an etching process sequence to remove at least a portion of one of the plurality of fins and thereby define a fin cavity, wherein the etching process sequence includes performing a first anisotropic etching process and, after performing the first anisotropic etching process, performing a second isotropic etching process. In this embodiment, the method concludes with the step of forming additional insulating material in the fin cavity.