Yes it's possible for your throat to collapse.
When your throat collapses it's known as a condition called Tracheomalacia.
Tracheomalacia is a rare condition that occurs when the cartilage of the windpipe, or trachea, is soft, weak and floppy. This can cause the tracheal wall to collapse and block the airway, making it hard to breathe.
Shortness of breath is not usually enough to kill you so it's unlikely that you would die or suffocate from shortness of breath.
As long as you're breathing some and getting some air into your lungs then you're not likely to die but it's still not good if you're always suffering from shortness of breath.
If you constantly suffer from shortness of breath you should let your doctor know because you could have some lung issues or other health problems.
Heart disease can cause shortness of breath as well so it's best to get checked out.
It's sometimes normal to gasp for air if you've been exercising hard such as running and then you have to stop and catch your breath.
However you should not normally have to gasp for air to breathe.
If you suddenly have to gasp for air to breathe then you should seek medical attention especially if you are continuously having trouble breathing.
When breathing you should not have to gasp for air if your lungs are working properly and you should be able to breathe in and breathe out without even thinking about it.
When you're breathing normally you breathe in air and exhale hair thousands of times per day without even thinking about it.
You shouldn't have to struggle to breathe and if you do constantly have to struggle to breathe then something is not right.