Mechanical response during bending experiments of Ni-Mn-Ga-Co-Cu melt-spun ribbons with the L21 austenite structure was studied. This material exhibited anisotropy in mechanical properties depending on the side to which the applied bending force was directed. When force was applied to the “free side,” a substantial load drop was observed in the initial stage of bending. On the other hand, no load drop anomalies were observed when force was applied to the “wheel side.” Additionally, mechanical training effects were assessed by applying up to 10 bending cycles. It was demonstrated that with an increase in the number of bending cycles, there was no significant decrease in bending force, and the load–displacement curve remained unaltered. The temperature dependence of annealing of the ribbons' mechanical properties was also examined. The Ni-Mn-Ga-Co-Cu melt-spun ribbons were annealed at 373 K, 573 K, 773 K, 973 K, and 1173 K for 30 min, followed by subsequent bending tests. Annealing influenced bending response through two major phenomena detected. The first was associated with crystal structure ordering and recovery; while, the second was attributed to grain growth. Changes in mechanical properties influenced by different annealing temperatures were correlated with alterations in the microstructure of the studied ribbons.