The person that provided more than half the support has always been able to claim them. You're probably referring to the IRS Child Tax Credit. You get one thousand dollars per child that can be deducted from your tax liability but you don't get a check back if you have more credits than the liability.
To qualify, you have to meet the following criteria:
AGE: To qualify, a child must have been under age 17 – age 16 or younger – at the end of the year.
RELATIONSHIP: To claim a child for purposes of the Child Tax Credit, they must either be your son, daughter, stepchild, foster child, brother, sister, stepbrother, stepsister or a descendant of any of these individuals, which includes your grandchild, niece or nephew. An adopted child is always treated as your own child. An adopted child includes a child lawfully placed with you for legal adoption.
SUPPORT: In order to claim a child for this credit, the child must not have provided more than half of their own support.
DEPENDENT: You must claim the child as a dependent on your federal tax return.
CITIZENSHIP: To meet the citizenship test, the child must be a U.S. citizen, U.S. national, or U.S. resident alien.
RESIDENCE: The child must have lived with you for more than half of the tax year. There are some exceptions to the residence test, which can be found in IRS Publication 972, Child Tax Credit.