This is worst form of apology that a person can try to offer. It is the most insincere form I can think of. By using the word "but" after the I'm sorry, negates the whole apology all together. The stuff that follows the "but" is always justification for the wrong doing in the first place, therefore, the person is not truly sorry. If wanting to make your point and justify your actions, then just forgo the "I'm sorry" part. If there is part of the action or the reaction that you are truly sorry for, you need to use the form below and be intrinsically clear as to what it is that you are apologizing for. Then, that for which you are not sorry, may then be discussed and your point can then be made, independent of the apology.
I know we have all experienced this, have done this. It leaves the person that is being apologized to feeling even less important than if you had said nothing at all. Apology is something that should never be entered into lightly, since if you feel the need to make an apology, you have done something that has hurt another human being.