It can depend on the athlete. For example someone that is not expected to even place, coming in third can be a huge accomplishment. Often times these athletes end up doing amazing things they have never done before - last night in ice skating a women from Italy overcame previous failings where she was the first ever from her country to even medal in this event.
On the other hand you have the opposite happen...Some one that is expected to already be the best in the world and then fall short of even getting a medal.
For most though even making the Olympics is the huge accomplishment - anything over that is simply gravy. Most do the opposite of what you suggest....meaning to purposing fall short so as not to be third. Many, even though they may have no chance...try to do their own personal best. Doing their best, especially after a fall (or similar depending on the sport) actually show true character and a true athlete -- for these athletes giving up is not an option. They would never be where they are if they had that sort of attitude. It is called perseverance.
Many of these athletes that fall short --- if you can even consider having a bronze medal as falling short -- end up coming back in the next to win. If they start by their first try giving up, they will never reach the gold level.
And like others say -- to be third in the entire world, you think that is bad or failure -- not so much.