In my discipline (studio art) complimentary funding is encouraged, if not expected. There is a small fund available (roughly $20,000) through my college that is available for students to apply to for a variety of reasons. I applied this last month to the fund and maxed out the possible funding at $2,500 for supplies for my project, art supplies and money for things like a voice recorder. I was told by my professor that I was virtually guaranteed the funding due to the Fulbright nomination because it's so prestigious and rare in my department.
Now, I say all of this in regards to smaller grant opportunities. I suspect that applying to larger opportunities might be different if the proposal didn't clearly outline specific needs to be met by multiple grants of over $20,000. If you were to apply for a Fulbright after already receiving a 30 or $50,000 grant, I have a difficult time seeing them award a Fulbright as well, mostly because I have the sensation that Fulbright wants to help candidates get to where they want to go who wouldn't be able to get there otherwise. Of course, this is just my theory at this point.