No, only one person can get the Earned Income Credit (EIC). Regardless of your arrangements for claiming the children's exemptions, only the custodial parent can claim the EIC (along with Head of Household and the Child and Dependent Care Credit). The IRS determines custody based on who the child spent the most nights with during the year.
If you indicate 7 or more months in the Personal Info section for your child to have been living with you, then TurboTax would have made you the custodial parent and given you what you were eligible for. If you indicated the other parent would be claiming, you would not have gotten the $4,050 reduction of income for them or the Child Tax Credit.
If the other parent is the custodial parent and he is claiming the children for the entire year (or more than six months), then you should leave them off your return. You can delete a dependent from the Personal Info section on TurboTax.
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