Yes you can refinance your conventional mortgage into a FHA mortgage. 

FHA will allow you to refinance to 96.5% loan to value but there can be no cash out.  Loan to value means the amount of your loan as a percentage to the value:  example value of home is $200K with a loan of $193K gives you a loan to value of 96.5%.

If you are trying to refinance an into an FHA loan and taking cash out than there are limitations.  Most banks will not allow you to take out cash beyond 85% loan to value.  What does that mean:  If your home is worth $200K you cannot have an FHA loan with cash out beyond $170K.

FHA has max loan amountsallowed that are based by state and than by county.  These limits are set by HUD.  The max loan limits are set by HUD and are 120% of the median home price for the county to a max loan amount of $729,750.  Example:  Calaveras County, CA max loan amount is $462,500 for a single family home versus in San Francisco County, CA the max loan amount for a single family home is $729,750 which is the max amount HUD will allow for FHA. 

Call a Loan Professional to talk about your circumstances to see if an FHA loan is the right approach for your situation.

Of course it's possible. In fact, many people are taking that route because their equity position isn't strong enough to allow them to be approved by a conventional lender. FHA loans do require a Mortgage Insurance Premium (MIP), which may be higher than the standard private mortgage insurance charged by conventional lenders. However, the upfront MIP can be financed into your FHA loan, so you don't have to pay it out of pocket.

In addition, FHA does not impose the surcharges that Fannie Mae and Freddie Mac do as part of their risk-based pricing. A look at Fannie Mae's Loan Level Pricing Adjustmrnt matrix shows you how easily you could end up with 3, 4, or more points tacked on to your loan fee, especially if your credit score is low or you don't have lots of equity. FHA may well be the best choice for you.

Finally, FHA underwriting guidelines are a little more forgiving than those of conventional loans. There is no automatic credit score requirement; borrowers applications are reviewed individually, and if you can prove that past credit problems have been resolved and are unlikely to recur, you have an excellent chance of getting a loan. In addition, FHA underwriters are allowed more latitude regarding income--for example, if your proposed housing payment is a little high for your income, but you have been successfully paying a similar amount in rent for some time, you have a great chance of getting approved.

FHA loans have become very popular for a number of reasons. But even FHA has been slowly tightening up its underwriting guidelines and appraisal requirements, and rates aren't getting any lower. I suggest that you shop for a good lender and start the process very soon.

Since you will be paying off the original conventional mortgage note, you are able to finance into any new mortgage program you choose. FHA mortgages require an upfront mortgage insurance premium or "MIP" of 1.75% that generally is financed into the new loan amount. In most cases FHA mortgages will also require annual mortgage insurance paid monthly for at least 5 years regardless of the amount of equity you currently have into the property. The exception being a 15 year term where no annual mortgage insurance is required if the loan is less than 90% of the current appraised value or otherwise known as less than 90 "LTV" (Loan To Value).  If your loan amount will be less than 80% of the current appraised value, your best option might not be FHA. The short answer to your question is yes, you can refinance your conventional loan into an FHA loan. My recommendation would be to consult with a mortgage professional who can assess your unique situation and present you with your best possible options.

Yes, FHA can be used to pay off a conventional loan. Here is an example to where that may be beneficial. 

I recently had a client who could not use the government refi program that allows for loan to values in excess of 90%.  She was recently divorced and since the original borrowers are required to be on the new loan in those situations, she was ineligible.  So we were able to refi her home using FHA since the loan to value was at 96.5%. There are other loans that were not owned by Fannie Mae or Freddie Mac that could be paid off using an FHA as well. 

In general however, if conventional financing is available, it is typically the most prudent source of funds.