Yes, but not as much as you might imagine. For one thing, we bill based on your usage each day. For instance, if your site gets aggressively crawled by many bots, that usage would only be reflected in that day's costs. A single day of heavy usage is not a huge expense; it does not drive up your costs for the entire month.
It's important to keep in mind that your website will suffer poor performance if the same thing happens on a limited server. Our customers value their applications running smoothly, and that is a worthwhile compromise.
You can learn more about how our billing works on this page.