Q. Can I claim them as other dependents if they filed also?
A. No. A person can only be claimed on one tax return.
As Opus17 explains, it's possible for you to claim them, depending on the details, and requiring them to adjust their filing(s). It is highly unlikely that would work out best for the family. Doing practice returns and comparing the results, before actually filing, is highly recommended. The grandchild would most likely qualify as a "qualifying child" rather than an "other" dependent.
