Was approved within a couple hours. Signed the contract the next day. I've heard some horror stories about getting an apartment in DC but mine was approved within a couple hours. I'm in the heart of DC, literally a couple blocks from the white house.
I think I was incredibly lucky. Since I'm not a citizen I was worried. In Sydney there's a huge waiting list like 30 peeps for a nice place and you need to know the real estate agent personally to get an affordable deal, also need to show income and other things. Here, it's just a telephone interview, fill out forms, background check, that's it. Simple.