Is it possible to improve one's taste palette through practice or other methods?
Sure...well, almost.  I don't believe you will be training your actual taste sensors to become more sensitive (biologists or flavor chemists can correct me), but you can learn to improve your recognition and understanding of flavor. So, it is a matter of training your brain, not your tongue.  It's how chef's, sommeiliers, food/taste chemists, and even home cooks get better and become experts.  Taste a lot...observe...take notes...practice.  I would start with one food, one cuisine, or one beverage (like wine), and learn as much as possible about it, by which I mainly mean lots of tasting. You will find that you begin to recognize aroma and flavor profiles, variation and even imperfections.  Of course, this takes time, but the journey can be very rewarding.