Depends - if your profits increase in line with your expenses, you might be raking in the ad revenue/views/sales/whatever because your product 'went viral' and you might prefer to keep it up.

Or if the rise in costs is something you can mitigate on your end - such as a bad deployment - you might want time to respond yourself, rather than your site going offline.

More generally, few site reliability engineers are looking to add extra ways for the site to be taken offline.

Of course, if you're large enough to be in those situations, your round-the-clock operations staff will be monitoring the billing situation as carefully as they monitor page load times and error rates and database load so an unexpected bill will be very unlikely.