No.
A cousin, even a first cousin, is not considered related for tax purposes. So, would have to live with you all year to qualify as a dependent. of course you would also have to have provided more than half her support. There's also the issue of her probably being the qualifying child of another taxpayer.
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.
A person can still be a Qualifying relative dependent, if not a Qualifying Child, if he meets the 6 tests for claiming a dependent:
1. Closely Related OR live with the taxpayer ALL year
2. His/her gross taxable income for the year must be less than $4,050 (2016)
3. The taxpayer must have provided more than 1/2 his support
In either case:
4. He must be a US citizen or resident of the US, Canada or Mexico
5. He must not file a joint return with his spouse or be claiming a dependent of his own
6. He must not be the qualifying child of another taxpayer