Could someone feel his or her heart flutter?

Yes: Heart palpitations are sensations of fluttering or skipped heart beats. Atrial flutter is a rhythm disorder which can often be sensed by people who have it.

Palpitations: Palpitations have many causes from anxiety through anatomic and physiologic abnormalities. The feeling of bounding usually relates to the volume of blood pumped with each palpitation. There is no simple single answer for the question. If you're bothered by the feeling should consult a dr. To see if the problem can be clarified.