The first time he is sent back, he ends up in the year 1990, where he is promptly put in an asylum. The plot is set up so it looks as though Cole is responsible for Doomsday. Goines then goes on to become the leader of the Army of the 12 Monkeys, who the people in 2035 believe are the ones responsible for unleasing the virus. Meanwhile, the scientists in Cole's future bring him back and have another go at putting him in the right year. However, Reilly's psychiatry and the continuous time travel appears to have taken its toll on Cole, and when he is brought back to 2035 and sent back to 1996 again, he is convinced that he is just a mental patient with delusions about the future.
History can lie.
At this point, there are numerous options to end the story after tieing up the lose ends. But the story is left very open-ended and the issue of whether there are more loops in the future and the question whether the past can really be changed is left unexplained and unanswered. The Next Generation episode where the Enterprise is caught is a time loop of destruction, and each time the crew travels back through time, they experience a deja vu sensation that finally allows them to break out of the destructive loop. The memory Kathyrn Reilly experiences, that she has seen James Cole before, could be one such instance. Thus viewers who like happy endings may wish to imagine another loop where Reilly and Cole actually manage to save the world.