"weak chest" is correct and usable in written English.
You can use it to describe someone who has difficulty breathing or is having a hard time filling their lungs. For example, "The patient had a weak chest and was struggling to take a deep breath."
As a young man Lawrence had been proud of his body, his "weak chest" notwithstanding.
Descartes had been a puny child with a weak chest and was not expected to live.
This body-proud countryman with "a weak chest" became the man whose rotting body filled him with miserable self-loathing.
He had always had "a weak chest": in those days this was often a euphemism for tuberculosis.
Due to a condition referred to as a "weak chest", Karl was excused from military duty when he turned 18.
All four of Adolph's sons were subject to what one biographer calls "benign neglect", and Gustav in particular was "not overburdened with attention or understanding, with a weak sight and a weak chest, both neglected he was 'miserable and scared'".
It was a place where customers who suffered from delicate dispositions and "weak chests" would literally be "restored".
These locations were meant more as meal replacements for those who had "lost their appetites and suffered from jaded palates and weak chests".
"Weak chests, you know, um in my family, you know, most of my family have, you know, had asthma….
Likewise, respondents were asked if the following were warning signs of a heart attack: pain or discomfort in the jaw, neck, or back; feeling weak, lightheaded, or faint; chest pain or discomfort; pain or discomfort in the arms or shoulders; shortness of breath.
And Google recently bought over 1,000 patents from I.B.M. covering multiple areas of technology to bolster its relatively weak war chest.
