There is no such rule. What you may have heard of is a written analysis of what the current rules actually say. There are probably many of them. The custodial parent has first priority on claiming the children on her taxes; regardless of the amount of support provided by the non-custodial parent. The non-custodial parent can only claim the child as a dependent if the custodial parent gives permission (on form 8332) or if it's spelled out in a pre 2009 divorce decree. Even if a divorce decree, dated after 2008, gives the non-custodial parent the right to claim the child, he must still get form 8332 from the custodial parent. A properly worded decree should require her to provide that form.
If there is a court approved divorce agreement giving the noncustodial parent the right to claim the child, and that clause is not dependent on the payment of child support, then he can still claim the child, under that agreement. BUT the IRS will not enforce a post 2008 divorce decree. The non-custodial parent's only remedy, if the custodial parent claims the child, is to take her back to court.
Here's something else, you should be aware of: There is a special rule in the case of divorced & separated (including never married) parents. When the non-custodial parent is claiming the child as a dependent/exemption/child tax credit; the custodial parent is still allowed to claim the same child for Earned Income Credit, Head of Household filing status, and day care credit. This "splitting of the child" is not available to parents who lived together at any time during the last 6 months of the year; then only one of you can claim the child for any tax reasons. The tax benefits may not be split in any other manner.
Note in particular that
the non-custodial parent can never claim the Earned Income Credit, based on that child.