FORT LAUDERDALE, Florida (CNN) -- Francisco Rodriguez owes more than $10,000 in back child support payments in a paternity case involving a 15-year-old girl who, according to DNA results and the girl's mother, is not his daughter.

Francisco Rodriguez is fighting for leniency in his paternity case. "It's not right. I'm not the father, " he said.

Rodriguez, who is married with two daughters and a son from his wife's previous marriage, is fighting for leniency. "It's not right. I'm not the father, " he said at a recent court hearing.

He says he knew nothing about the other girl until paperwork showed up about four years ago saying he was the father.

He now has DNA results that show the 15-year-old girl wasn't fathered by him. He even has an affidavit from the girl's mother -- a former girlfriend from 1990 -- saying he's "not the father" and asking that Rodriguez no longer be required to pay child support.

Yet the state of Florida is continuing to push him to pay $305 a month to support the girl, as well as the more than $10,000 already owed. He spent a night in jail because of his delinquent payments.

Why is he in such a bind?

He missed the deadline to legally contest paternity. That's because, he says, the paperwork didn't reach him until after the deadline had passed. Watch Rodriguez plead in court for a break »

"It's like you're drowning every day," says Rodriguez, a massage therapist.

Rodriguez's case highlights the legal dilemma states face over how to handle paternity cases. More than a third of children born in the United States are born to unmarried parents, according to the National Center for Health Statistics.

But paternity laws vary from state to state, according to the Center for Law and Social Policy (CLASP), a nonprofit organization that works to improve the lives of low-income families.

Supporting a stranger "Paula Zahn Now" examines men paying child support for children that aren't theirs.

Thursday, 8 p.m. ET see full schedule »

Some states have detailed laws to challenge paternity within deadlines, while others offer little guidance. In most cases, men have 60 days to challenge paternity, according to CLASP.

After that, it can be "challenged only on the basis of fraud, duress or material mistake of fact," CLASP said last year in an update to a report on paternity law.

"There are no perfect answers," says Susan Paikin of the Center for Support of Families in Delaware. "Deadlines are imposed so that when families are broken -- the legal process is handled quickly."

She says state legislatures and courts struggle with paternity cases, trying to strike the proper balance between children's rights and adults' rights, always keeping in mind any potential harm to the child.

"This is a struggle. It's not something easy for courts or legislatures," she says.

Paikin says it's especially tricky in cases where a father has raised a child thinking it was his, only to learn years later the child had a different father.

"Most men who have a relationship with their child don't think of their child in terms of DNA," she said. "The real issue in most of these cases is anger and money."

Tampa Police officer Michael Anderson understands that sentiment. He paid child support for more than 12 years -- a total outlay he says amounted to more than $80,000. But a DNA test after he and his wife divorced showed the daughter he thought was his was somebody else's.

He then separated himself and his feelings from the child.

"I stopped having a relationship with the girl right from the beginning, when I found out," he said. "It was hard, but I had to do it."

A Tampa court earlier this year disestablished him as father and relieved him of his future child support payments. But by law, he is unable to get back the $80,000 he already paid.

Carnell Smith, who founded a group called U.S. Citizens Against Paternity Fraud, wants mandatory DNA tests when a child is born to avoid legal wrangling and anguish.

"Unfortunately, today it's not a crime for someone to lie about which man is the father," Smith said. "The mother doesn't have to return the money and rarely, if ever, is she prosecuted for perjury, for fraud."

Rodriguez's odyssey began in 1990, when he says at age 16 he had a four- to five-month relationship with a woman CNN is not identifying. He says when the relationship ended, he did not hear from her again until child support papers arrived at his home in 2003.

"My wife and I both had a confused look, and we're wondering, 'Where is the DNA test?' " he says.

But it was long past Florida's deadline to contest paternity. A court had already named him the father three years before when he did not respond to notices to appear, notices he says he never received because he had moved a lot.

He was now on the hook for monthly child support, as well as $10,623 in back child support.

He eventually paid for DNA testing. The test showed he was not the father.

A judge has now ordered a court-sanctioned DNA test for Rodriguez and the 15-year-old girl. Rodriguez has taken that test; the girl and her mother did not show up for their appointment to submit to DNA testing and it's unclear if the girl has complied.

CNN has repeatedly tried to contact the mother, but has been unable to reach her.

Rodriguez and his family continue to wait for answers.

"It's hard when your daughter needs sneakers and you have to pay $305 or your husband goes to jail," said Rodriguez's wife, Michele. "It's just unfair." E-mail to a friend

CNN's Rich Phillips contributed to this report.

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