It is feasible to go for a refinance when you have built up at least 10% equity in your home (For Fannie Mae owned mortgages, the value is 5%). It is also possible for you to choose the option if your equity is less than 5%, but you may have to pay a certain amount of cash in order to make up for the difference in equity.
When refinancing, don't take the first offer that comes around. Instead you should actively compare both interest rates and lenders.
If you are able to get a lower rate that what you currently have, you can save tens of thousands of dollars over the life of your loan. Also, most lenders don't charge as many fees to refinance a mortgage and depending on how much equity you have in your home you may be able to roll the closing costs into your new loan, still have a lower balance than your original loan, a lower rate, and a lower payment.
There is a refinancing myth that says you should not refinance your mortgage unless your interest rate will be at least two points less. This myth is not necessarily true if there are other benefits to the refinance or other reasons behind it.
Ask the lender or broker for a comparison of the up-front costs, principal, rate, and payments with and without this rate trade-off.