Michael Luo of the New York Times discusses the various attempts by Barack Obama's opponents to test just how easy it is to donate to the Obama campaign using fake names and addresses ("Donor Patrol: ObamaaMichael Luo of the New York Times discusses the various attempts by Barack Obama's opponents to test just how easy it is to donate to the Obama campaign using fake names and addresses ("Donor Patrol: Obama’s Online Site Accepts More Fakes", link). He spins the story completely in favor of the Obama campaign to the point of covering up for them:

[The fake donation] experiment would not necessarily be notable, except it appears that many others are doing the same thing. Power Line, a conservative political blog, reported a reader had successfully made donations under the names Osama bin Laden, Saddam Hussein and Bill Ayers. While those experiences could not be immediately verified, Ms. Franzi sent The Caucus a screenshot from her bank account that showed a contribution to Obama for America going through at 1:02 p.m.

To be fair to the Obama campaign, officials there have said much of their checking for fraud occurs after the transactions have already occurred. When they find something wrong, they then refund the amount...

However, unless the Obama campaign is given the full credit card number and is storing it (a huge security risk considering they don't have the same expertise and experience as processing companies), they're unable to later verify that a specific credit card number belongs to a specific person at a specific address. That's a bit like a store checking for fraudulent currency not at checkout but after dozens of bills from different parties are already in the cash drawer. See also this.