To begin with, the IRS does not care about court orders or agreements when determining dependency status. All that matters is the tax law.
While the rules for claiming a dependent can be complex, here are the basic requirements to be able to claim a qualifying child as a dependent - you can see more information at these links Rules for Claiming a Dependent on Your Tax Return and IRS Publication 501 (starts at page 11).
To be a Qualifying Child -
1. The child must be your son, daughter, stepchild, foster child, brother, sister, half brother, half sister, stepbrother, stepsister, or a descendant of any of them.
2. The child must be (a) under age 19 at the end of the year, (b) under age 24 at the end of the year and a full-time student or (c) any age and permanently and totally disabled.
3. The child must have lived with you for more than half of the year. Temporary absences while away at college are considered living with you.
4. The child must not have provided more than half of his or her own support for the year.
5. If the child meets the rules to be a qualifying child of more than one person, you must be the person entitled to claim the child as a qualifying child.
6. The child must be a U.S. citizen or U.S., Canada or Mexico resident for some portion of the year.
7. The child must be younger than you unless disabled.
From the information you have presented, it appears that the child would qualify as your boyfriends dependent as the child was with him more than half the year and he likely provided more than half of her support. I am going to assume that the child meets all the other requirements, but you will need to double check all of them. And, when you answer the questions in TurboTax, the program will determine eligibility.
Now, the big problem here is if the other party already claimed the child (even if they should not have) and their return gets processed first by the IRS, then the e-file system will not allow him to also e-file his return. So, he would have to paper file his return in this scenario.
But, he would still be able to claim the child on his return and then the IRS will decide who is truly eligible for the dependent and related credits/deductions. So, while the process might get a bit involved, it is possible for him to still claim the child if he believes (and, most importantly, if he can prove) that he meets the qualifications to claim the child as a dependent.
