"I made around 6,000 last year"  Was any of that cash or self employment income? If so, you may be required to file a return, which could change the answer below.

Your boyfriend cannot claim you because you made more than $4050. But he can claim the kids, because you made less than $10,400*. 

BUT, your child may qualify as a dependent, but because he is not related, he cannot be your BF's qualifying child for the earned income credit, child tax credit or Head of Household filing status.

 There are two types of dependents, "Qualifying Children"(QC) and standard ("Qualifying Relative" in IRS parlance even though they don't have to actually be related). There is no income limit for a QC but there is an age limit, a relationship test and a residence test. Only a QC qualifies a taxpayer for the Earned Income Credit and the Child Tax Credit.

 A person can still be a Qualifying relative dependent, if not a Qualifying Child, if he meets the 6 tests for claiming a dependent:

1. Closely Related OR live with the taxpayer ALL year

2. His/her gross taxable income for the year must be less than $4,050 (2016-17)

3. The taxpayer must have provided more than 1/2 his support

In either case:

4. He must be a US citizen or resident of the US, Canada or Mexico

5. He must not file a joint return with his spouse or be claiming a dependent of his own

6. He must not be the qualifying child of another taxpayer*

*In addition to the above requirements, to claim your boy/girlfriend's children, they must meet all of the above requirements and:
--- your boy/girlfriend must not be required to file a return,
--- he/she does not file a return claiming the children

So, if your $6000 of income was from self employment, and not W-2 wages, you are required to file a return and your BF cannot claim your kids

You should consider "making a deal" with the ex-husband to let him claim the children He cannot claim the EIC or HoH, because the children do not live with him, but he can claim the Child Tax Credit and the children's exemptions ($4050 deduction)
Furthermore, you would be able to claim the children for the EIC, on your return (about $2400 on $6000 of earned income). You do not need to let the ex claim the kids to get EIC; but you get little more for claiming them. You will get about $450 of child tax credit whereas the ex will get about $3000, or more.
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 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, Head of Household filing status or the day care credit, based on that child, even when the custodial parent has released the exemption to him.
 So, it's good idea to let the other parent know that you will be claiming those items, as many first time divorced parents are not aware of this rule and may try to claim those items, which will cause the IRS to send out letters.
Ref: <a rel="nofollow" target="_blank" href="http://www.irs.gov/publications/p17/ch03.html#en_US_2014_publink1000170897">http://www.irs.gov/publi...> Scroll down to "Children of divorced or separated parents (or parents who live apart)"