It is unfortunate that you cannot afford an attorney because there are bigger issues at work here. The short answer is probably not, and certainly not in family court. To be clear, if you are opposing a finding that you are this child's parent for purposes of custody and support, you have no grounds to initiate an action to establish paternity just to say you are not the parent and you want a refund. Also, the court does not merely look at biological relationships to establish paternity and you may have opened yourself up to some risk despite your paternity test. For example, you will want to be sure that you are not on the child's birth certificate or have not signed a voluntary declaration of paternity, in which case you may have to take further steps to establish that you are not this child's father.
To the extent the mother has coerced you into paying for her birth under the false representation that the child is yours - you may have claims for a civil suit or small claims action, but depending on how much money we are talking about, and whether the mother was even aware that you were not the father, it may not be worth pursuing.
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It probably depends on a particular factual circumstances that exist. If you came to your own conclusion you were the father and you insisted on paying for the birthing, you probably will not receive reimbursement. However, if she swore to you it was your child and begged you to pay for the medical care, then I believe you may have an unjust enrichment situation, and she should be obligated to reimburse you. An analogy would be sort of like: She promises to marry you, you buy the diamond ring, then she tells you to go jump off a bridge. Under those circumstances, you would probably get the ring back; at least in the State of California. But, if in fact she were to marry you, then the ring is hers.
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