Can I claim my nephew even though my sister puts him down as a dependent on her financial aid?

Yes, as far as the IRS is concerned. A child closely related to a taxpayer can  be a “Qualifying Child (QC)” dependent, regardless of the child's 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

3.                   He lived with the relative (including temporary absences) for more than half the year

4.                   He is younger than the relative (not applicable for a disabled child)

5.                   If the child meets the rules to be a qualifying child of more than one person, you must be the person entitled to claim the child as a qualifying child (this essentially means that you have the parent’s permission to claim the child)

6.                   If the parents of a child can claim the child as a qualifying child but no parent so claims the child, no one else can claim the child as a qualifying child unless that person's adjusted gross income (AGI) is higher than the highest AGI of any of the child's parents who can claim the child.         

You'll have to ask the financial aid people about how that will affect his aid.