How can I log in with a different account

I need to log in with different accounts in a web site. I did that for a long time. But now, they changed and added a new login window on top and whenever I log in with a different username, it's as if I'm logged in as the first username I used.
I tried to clean my cookies or change my IP but there was the same result.
Then I waited the next day and I could log in successfully with a different account.
The problem is that I don't want the next day for every account.
Is there a trick that would allow me to log in with a different account when I want?

Use both firefox and google chrome.
Or any 2 browsers.

Works for me.

they probably do some IP lock or something , try changing your IP and logging in differently.

Cause clearing cache, cookies should work just fine for switching accounts.

Try using private mode in firefox, or incongnito mode in chrome.

try to clean Flash cookies at \Application Data\Macromedia\Flash Player\#SharedObjects\
. but if they check IP you have to change Ip each time.

Some sites are smart and track your IP on a time limit base I think. So it might be a daily check of your IP and after you crossed the 1 day limit,your IP was not tracked.