Recent advances in quadrupedal robots have demonstrated impressive agility and the ability to traverse diverse terrains. However, hardware issues, such as motor overheating or joint locking, may occur during long-distance walking or traversing through rough terrains leading to locomotion failures. Although several studies have proposed fault-tolerant control methods for quadrupedal robots, there are still challenges in traversing unstructured terrains. In this paper, we propose DreamFLEX, a robust fault-tolerant locomotion controller that enables a quadrupedal robot to traverse complex environments even under joint failure conditions. DreamFLEX integrates an explicit failure estimation and modulation network that jointly estimates the robot's joint fault vector and utilizes this information to adapt the locomotion pattern to faulty conditions in real-time, enabling quadrupedal robots to maintain stability and performance in rough terrains. Experimental results demonstrate that DreamFLEX outperforms existing methods in both simulation and real-world scenarios, effectively managing hardware failures while maintaining robust locomotion performance.