Ideas don't belong to anyone, anyway.
Can you truly get behind an idea that's not your own... if you have another reason for being there?
Yes.
I want to start my own company (based on my own idea) some day, but I'm learning hell of a lot (and saving up cash) working at another company, on someone else's idea. Getting behind that idea, even if I'm not necessarily passionate about it, is important, because that's the only way I'll do a good job and actually learn what it's like.
So yeah. I think it's important to gain some experience before setting out on your own. It's a safer route if you don't have people to fall back on if your first (or second or third) attempt fails
As an extreme example, if I were passionately committed to solving world hunger and the guy running the start-up I worked for found a solution, commitment to that idea would be possible.
Ideas are about Strategy - and Vision / Personal Intent are more powerful than Strategy.
Look at how many religious people they are - they all emphatically support an idea that was not their own.
As long as you enjoy working on it why not? Besides, once you start working, the idea changes anyway. Very few ideas stay true to form the whole way through. I think the problems arrive when the people "behind" the idea are completely unwilling to change it at all.
Would anyone really say no? Maybe you meant to ask a more specific question?
I love cool ideas. But I really love the cool ideas I'm executing. So execution counts more than initial infatuation, even though it doesn't always seem so before you get started.