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Drowning in student debt? Maybe you can refinance

Millions of Americans could put more money in their pockets each month by refinancing their student loans. It's always been possible to refinance private student loans, but few financial institutions offered this service.

In January, RBS Citizens Financial Group, which operates Citizens and Charter One banks, introduced a new Education Refinance Loan, with fixed rates as low as 4.75 percent APR and variable rates starting at 2.8 percent APR. There are no application, origination or disbursement fees and you don't have to be a bank customer to apply for this refinance.

"These are very aggressively priced and typically a much better rate than a loan you can take out when you're in school," said Brendan Coughlin, head of education and auto finance at RBS Citizens Financial. "For a vast number of students who have private student loan debt, this will dramatically improve their payments and the interest they pay over the life of their loans."

(Read more: Value of college? For millennials, $17,500 a year)

Coughlin said the company sees this as a very big business opportunity and a way to build lifelong relationships with students and their families.

Citizens and Charter One will only refinance private loans. SoFi, a student loan company based in San Francisco, will also refinance government loans. SoFi has refinanced $400 million in student loans since 2012.

"And that's just the tip of the iceberg," said SoFi's CEO, Mike Cagney. "There are hundreds of billions of student loans out there that have a great potential to be refinanced and we hope to get the word out."

Cagney told me the average customer saves between $5,000 and $9,000 over the life of the loan.

Gil Eyal is glad he heard about SoFi. After getting his MBA from Northwestern University, Eyal moved to Hoboken, N.J., and worked on a social media start-up. He refinanced $100,000 in federal loans with SoFi—from 8.5 percent to 6 percent—and cut his monthly payment by $500.

"We were barely breaking even," Eyal said. "Now, we can save some money every month, which definitely gives us breathing room and the ability to have the paycheck go a little further."

Are you a candidate?

Refinancing isn't for everyone. Basically, you need to be in better financial shape than you were when you took out the loans.

"You have to be employed and you're in the best position if your credit score is good, particularly if it has improved since those loans were taken out," said Greg McBride, chief financial analyst with Bankrate.com.

One of the most important things is to make sure that you're actually getting a lower interest rate.

"Your interest rate should go down, not just your monthly payment," said Rohit Chopra, student loan ombudsman and assistant director at the Consumer Financial Protection Bureau. "You might be able to get a lower payment simply by extending the term of your existing loan, but the interest rate could be higher which will cost you more in the long run. The way to save on interest is to make sure you get a lower APR."

Something else to consider: When you refinance a federal loan into a private student loan, you may lose some valuable benefits—such as public service loan forgiveness or the option to use income-based repayment if you run into trouble.

(Read more: Ready to pop: $1.2 trillion student loan bubble)

"Honest lenders will inform you about the benefits you might be giving up, but you also have to look out for yourself, too." Chopra said.

Know the terms

It's also important to understand the difference between consolidation and refinancing. Consolidation means taking a group of loans and rolling them into a single loan to make one payment each month. That's convenient, but you won't always get a lower interest rate.

"For some borrowers, those capable of accelerating payment on some of their loans, it may be better not to consolidate," said Mark Kantrowitz, a nationally recognized expert on student loans and publisher of Edvisors Network. "Your best bet may be to only refinance the loans with the highest interest rates and don't touch the ones with low rates. That way you can pay off the highest interest rate loans quickly and have a lower average interest rate than you would have if you consolidated."

(Read more: The millennials' rut: Why it costs all of us)

And names can be confusing. For instance, with the Wells Fargo Private Consolidation Loan, you can consolidate several loans or refinance a single loan.

There's a lot to think about before you refinance a student loan. The Consumer Financial Protection Bureau has some helpful information on its website:

—By CNBC contributor Herb Weisbaum. Follow him on Facebook and Twitter @TheConsumerman or visit The ConsumerMan website.

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