If you are on a shared/sdx server, you have a daily bandwidth limit. If you run out of daily limit, your website will be disabled temporarily till midnight. No automatic overage charges. To reopen the site before that you will need to upgrade your account to higher plan that gives you more bandwidth per day.
If you run out of your monthly limit, your account website will be disabled unless you purchase additional bandwidth in advance (this happens rarely because the daily bandwidth limit will not let you go beyond monthly limit unless you are using more per day before your site is disabled).
In both cases, you need to consider upgrading your account to a higher plan. The upgrade path for bandwidth usage is:
shared :: sdx :: vps :: dedicated server
Note: VPS and Dedicated Servers do not have any daily limit.