Not if it's the same child, and the people are filing a return for each person. In order to claim a child as an exemption you must include the child's Social Security Number on the return. I assure you that the IRS data will catch it if two people claim the same child as an exemption.
No, they can't. Even if they let you at the time of filing, the person who did it last will be said to have committed fraud and would have to actually repay that money.