I'd accept a reason that meant that the person was genuinely unable to make it *and* was unable to contact me.
This would include accident or injury to themselves or someone nearby - for example, if they came across a car accident administering first aid and helping the victims takes precendence over phoning me, as would going to the emergency room or taking someone there.
Not everyone has a cell phone, and batteries die and reception fades out (I've run into problems involving traffic jams and mountains before), and there are places where you can't make cellphone calls - church services, during meetings at work, at concerts, where you can't make a call to say that you're running late.
If odd accidents happen more than once, then I am less likely to believe them, and the matter of the apology counts.
On a similar note, I find that some people think that phoning to say that they're going to be late is just as good as arriving on time - I hate it when I've arrived at a meeting place only to get a cell call from somebody who's just managing to leave their apartment - I have better things to do that wait for them to get their act together. I have a theory that people in general are better at making sure that they're on time when there is no way of contacting the others.