No one has mentioned the case of commutative algebra. The appearance of computer algebra systems allowed researchers in commutative algebra and/or algebraic geometry to generate lots of examples which then lead to conjectures, later proved by hand. Look at this quote from Eisenbud's home page, for instance:
"Ever since the early 70s I've used computers to produce examples in algebraic geometry and commutative algebra, and I've developed algorithms to extend the power of computation in this area. I recently joined Mike Stillman and Dan Grayson in the project to (further) develop the Macaulay2 system for symbolic computation. "
