PITTSBURGH—Carnegie Mellon University is scrambling to recover after Uber Technologies Inc. poached 40 of its researchers and scientists earlier this year, a raid that left one of the world’s top robotics research institutions in a crisis.

In February, Carnegie Mellon and Uber trumpeted a strategic partnership in which the school would “work closely” with the ride-hailing service to develop driverless-car technology.

Behind the scenes, the tie-up was more combative than collaborative.

Uber envisions autonomous cars that could someday replace its tens of thousands of contract drivers. With virtually no in-house capability, the San Francisco company went to the one place with enough talent to build a team instantly: Carnegie Mellon’s National Robotics Engineering Center, or NREC.

Flush with cash after raising more than $5 billion from investors, Uber offered some scientists bonuses of hundreds of thousands of dollars and a doubling of salaries to staff the company’s new tech center in Pittsburgh, according to one researcher at NREC. Uber declined to comment on that.