I was recently at this workshop at Fields on 'discovery and experimentation in number theory': you can get audio and slides from many of the presentations here although the one I wanted to recommend - David Bailey's talk - appears to be audio only. It depends what you mean by 'major mathematical advance' but there certainly seem to be many problems in number theory where even if the proof doesn't depend on heavy computation, gaining insight into what to set about proving did. I personally logged several CPUweeks trying to get to grips with my own thesis topic.