WASHINGTON — Anthony Cooper shot a woman in Detroit in 2003 and then received laughably bad legal advice. Because all four of his bullets had struck the victim below her waist, his lawyer said, Mr. Cooper could not be convicted of assault with intent to murder.

Based on that advice, Mr. Cooper rejected a plea bargain that called for a sentence of four to seven years. He was convicted, and is serving 15 to 30 years.

At least Mr. Cooper heard about his plea offer. Galin E. Frye’s lawyer never told him that prosecutors in Missouri were willing to let him plead guilty to a misdemeanor and serve 90 days in prison for driving without a license. When Mr. Frye did plead guilty after the offer expired, he was sentenced to three years.

On Monday, the Supreme Court will hear arguments in the two cases, which ask how principles concerning bad legal work at trial should apply to plea bargains. The question is of surpassing importance, since a large majority of criminal cases are settled at the plea stage.