If you are claiming educational expenses, then no because because a high school isn't an eligible educational institution for claiming a credit. However, if you are asking this because you are 19 and still in school, you can be a dependent of your parents since you are still in school.

Let me know if this answers your question.

Yes, as long as graduation occurred in May or later.  For the dependency rule, the child must have been a full time student for parts of 5 calendar months.  The months do not need to be consecutive. 

There are two types of dependents, "Qualifying Children"(QC) and standard ("Qualifying Relative" in IRS parlance even though they don't have to actually be related). There is no income limit for a QC but there is an age limit, student status, a relationship test and residence test.

A child of a taxpayer can still be a “Qualifying Child” (QC) dependent, regardless of his/her income, if:

1. He is under age 19, or under 24 if a full time student for at least 5 months of the year, or is totally & permanently disabled
2. He did not provide more than 1/2 his own support. Scholarships are excluded from the support calculation
3. He lived with the parent (including temporary absences such as away at school) for more than half the year

Just to clarify:

because a high school isn't an eligible educational institution for claiming a credit

Determining dependent status and determining who claims any applicable education expenses are two different determinations.

For the purpose of determining dependent status, high school is qualified educational institution.

For the purpose of determining what education credits the tax filer may qualify for, high school is not an eligible educational institution.

This is basically covered in IRS publication 17 at https://www.irs.gov/pub/irs-pdf/p17.pdf on page 26 in Table 3.1 under the test to be a qualifying child dependent. Take note it does not define "a student" or the type of educational institution they are required to be in. Not even on page 27 under the "Age Test" criteria.