It might at some schools. In my discipline (math), anecdotal evidence suggests that MIT, Princeton and Stanford don't care about outside funding but Berkeley, Michigan and Austin do. I will spare you the plethora of stories I have heard about each, but I will elaborate on one example (let's take Berkeley) to show you what I consider anecdotal evidence.
A friend of mine got the NSF as an undergraduate. He notified Berkeley, which had initially rejected him. Supposedly they told him that they couldn't change the admission decision for that year, but they'd be happy to let him transfer in later provided that he didn't touch the NSF money in the meantime. I met another graduate student at Berkeley who transferred in after failing her qualifying exams at another institution. She too had 3 years of outside support. (I assume that Berkeley would have been much more cautious to admit a drop-out of another program if they had to fund her.)