You can only claim the American Opportunity Credit for payments you made in the year that you are claiming it for. On your 2016 tax return, you can only claim the credit for amounts that you paid in 2016. The billing date doesn't matter. It's when you actually paid that matters. However, the date of the classes matters, too. You can only claim the credit for amounts you paid for classes that start in the same year, or in the first 3 months of the following year. So if you paid at the end of 2015 for the spring 2016 semester, you can use that payment for the credit on your 2015 tax return. But amounts you paid in 2016 for classes that began in 2015 cannot be used for the credit at all, in any year.
