One illustrative method disclosed herein includes forming a plurality of initial fins in a substrate, wherein at least one of the initial fins is a to-be-removed fin, forming a material adjacent the initial fins, forming a fin removal masking layer above the plurality of initial fins, removing a desired portion of the at least one to-be-removed fin by: (a) performing a recess etching process on the material to remove a portion, but not all, of the material positioned adjacent the sidewalls of the at least one to-be-removed fin, (b) after performing the recess etching process, performing a fin recess etching process to remove a portion, but not all, of the at least one to be removed fin and (c) repeating steps (a) and (b) until the desired amount of the at least one to-be-removed fin is removed.