You're mixing up two separate sets of requirements: to be a dependent, and to claim the AOC.
To claim the AOC, your age & dependency status don't matter. But you have to be at least a half time student for at least one academic period beginning during the tax year. You also have to be enrolled in a degree program and not have claimed the credit in four previous years. The AOC is only available for the first four years of eduction, so it's unavailable to graduate students.
When you speak of age & "full-time" student status, you're talking about a dependency requirement. That's completely distinct from the AOC requirements. To be claimed as a Qualifying Child (one type of dependent) by a related person, you have to be under age 19 (or under age 24 if a full-time student) and you cannot provide over half your own support. There are also additional tests; see this link https://www.irs.gov/pub/irs-pdf/p17.pdf for all the requirements (see page 25).
Another type of dependent is the Qualifying Relative, which has its own set of requirements, including gross income. See the above link for those requirements, as well.
