Well, I just checked and the "Start Bounty" is ordinary submit button that is posting a form to the server with all the required information.
This means that if clicking it does not work some JavaScript code is blocking the click event.
Since there is no direct click event, most likely some jQuery code is attached at some point and on some browsers under specific conditions that still need to be reproduced, block the click event of the button.
As a workaround for those who can't start a bounty due to this problem, I suggest to disable JavaScript in their browser then the button will work just fine, being ordinary HTML submit button.
On second thought, disabling JavaScript won't work either since the bounty window itself is loaded by jQuery.