Has anyone won an olympic gold medal without also setting the fast time in the competition? For example someone else might have set the fastest time in the heats, but lost or been disqualified in the (slower) final.
In the men's 5000m, Mo Farah won with a time of 13:41.66, a time which would have placed him in the middle of the field in either of the heats. The fastest time was Dejen Gebremeskel at 13:15.15, 25s quicker than the winning time in the final.
Given how easy it was to find those (I just started with the track and field events at the most recent Olympics and worked up through the distances), I suspect this is a not uncommon occurrence, particularly in the longer distance events where races can be more tactical. In particular, the "fastest N losers qualify" rule will tend to force qualifying rounds to be quick to give athletes the best chance of making the finals, but there's no such pressure in the final.
I think that this happens in any sports in which athletes do the same timed sports event multiple times. In cross-country sprint the athletes have to follow the same course multiple times on the same day. Due to getting tired, their performances generally decline during the day.