If I cared about the person I was in a relationship with (which I currently do), I wouldn't even consider it. Actually, I was recently asked by an ex (like a week or two ago) to do just that, but I didn't even consider it.
Now, if I didn't care about the person, I probably wouldn't be in a relationship with them to begin with. But if somehow I was, I would end it before starting a new one. (Also, if the new one didn't work out, I certainly wouldn't come running back to the old one, like my last girlfriend did. Three times. *cough*)