Bismuth telluride (Bi₂Te₃) is one of the most important commercial thermoelectric materials. In recent years, the discovery of topologically protected surface states in Bi chalcogenides has paved the way for their application in nanoelectronics. Determination of the fracture toughness plays a crucial role for the potential application of topological insulators in flexible electronics and nanoelectromechanical devices. Using depth-sensing nanoindentation tests, we investigated for the first time the fracture toughness of bulk single crystals of Bi₂Te₃ topological insulators, grown using the Bridgman-Stockbarger method. Our results highlight one of the possible pitfalls of the technology based on topological insulators.