Yes - Interest you pay on student loans that you've refinanced or consolidated can still be deducted, as long as the new loan was used only to refinance qualified student loans.
To be deductible, the loan does not have to be with an official "student loan lender" and you can deduct interest even if you are not issued a 1098-E. (But, the IRS may send a letter asking for more proof.)
The purpose of the loan must be traceable to your education expenses. You could even deduct interest on a credit card if you used a credit card to pay qualified education expenses. But once you use the credit card for other things, you lose the ability to trace how much interest is due to the qualified education expenses and you lose the deduction. If you refinanced with a private lender, and used some of the loan proceeds to pay for other things besides the prior student loan, you are at risk of losing that ability to trace the interest dollars back to your qualified expenses.