Getting a mortgage refinanced is not all that hard to do, most people will qualify. While the banks have gotten more selective about who they will loan money to there are lots of lenders who will offer refinancing. You may have to shop around a bit but with one notable exception everybody should be able to refinance a mortgage.
The biggest exception where you may have a problem getting your mortgage refinanced is if your current mortgage is underwater. That means that you owe more on your existing mortgage than your house is worth. This is a situation that a lot of people have found themselves in recently as the housing market has tanked. In this case the bank is not going to give you a new mortgage since it won't be enough to pay off the old one. Even then however it may be possible to refinance, the government has created programs to help people who are struggling with their mortgage payments, some of these involve refinancing.
In almost all other situations you will be able to refinance, the bigger question then becomes should you? This is going to depend on why you are trying to refinance. If you are doing it to reduce the amount of interest that you have to pay, either by getting a lower rate or shortening the length of the mortgage then it usually makes sense. The interest on a mortgage will normally add up to be quite a bit more than the actual value of the house. Therefore any savings that you can get will be huge.
Another situation where it probably makes sense to refinance your mortgage is if you are not able to make the mortgage payments. In this case you can usually extend the length of the mortgage to make it easier to pay it. This will cost you a lot in interest over the long term but if you can't pay your bills there are not a lot of other options. That being said you do want to look at your budget and see if you can cut other things before you rush into refinancing.
Where refinancing becomes more controversial is when it comes to turning the equity that you have in your home into cash. There are times when it makes sense, like if you are going to use the money to pay off your high interest credit card debt. It makes a lot less sense if you are planning to use the money for a vacation. Remember you are borrowing that money and you are using your house as collateral for it. Make sure that what you use the money for makes sense from a financial point of view. The last thing that you want is to end up losing your house because you used the equity to take a vacation.