If men can work out and get a bigger chest can't a woman get a bigger chest with exercise too?

Yes: If you mean bigger pectoral muscles and other muscles, then yes. If you mean larger breast tissue. It does not work that way. The breast tissue is not muscle.

Sort of: Working out with the chest involves increasing the bulk and thickness of the pectoral muscles. This is possible with women also which might cause the breasts to protrude a little bit more, but not much. Check out women body builders to see what this can accomplish.