Originally a nautical term. When a sailing ship faced danger and needed to depart quickly, the rigging and anchor would be cut and the ship would run with the wind. More recently it has come to mean desert, retreat, or run away.
"When we were all out in the raw air and were steadily moving towards our business, I treasonably whispered to Joe, 'I hope, Joe, we shan't find them.' and Joe whispered to me, 'I'd give a shilling if they had cut and run, Pip."
Charles Dickens in Great Expectations
"The United States doesn't cut and run. When we make a commitment, we keep our commitments" President G.W. Bush (March, 2006)