When an actor reaches a point in their career where they have the opportunity to choose between several job offers, they run the risk of turning down the next big project.
Here are a few examples:
Leonardo DiCaprio was offered the role of Ennis Del Mar in "Brokeback Mountain," but thankfully he turned it down. As we now know, Heath Ledger got the part and was later nominated for an Oscar for his performance. Can you imagine "Brokeback" without Ledger?
That one must sting. Gwyneth Paltrow turned down the role of Rose in "Titanic" and never revealed her reasons. I can understand her. Then Kate Winslet came along and formed one of the most iconic on-screen duos in cinematic history with Leonardo DiCaprio. The film became a massive hit and garnered 11 of their 14 Oscar nominations. I'm glad Leo didn't turn down that film too...
Matt Damon had to turn down the lead role in "Avatar" because he was still filming the Bourne movies. Damon stated that he really wanted to play Jake Sully in Avatar but it was impossible due to scheduling conflicts. Avatar went on to become one of the highest-grossing films of all time.