They were once models of financial strength—corporate giants like AT&T Inc., Bayer AG and British American Tobacco Plc. Then came a decade of weak sales growth and rock-bottom interest rates, a dangerous cocktail that left many companies feeling like they had just one easy way to grow: by borrowing heaps of cash to buy competitors. The resulting acquisition binge left an unprecedented number of major corporations just a rung or two from junk credit ratings, bringing them closer to a designation that historically has made it much more expensive to fund daily business and harder to navigate economic downturns.