It may be possible for the most part, i think. Most of it depends on you though. How severe your symptoms are, what forms they take, ect. And whether or not you choose to take and stick with meds until you find a combo that works well for you with tolerable side effects.
I have several friends who are bipolar, i am bipolar and so is my boyfriend. My friends have taken meds in the past and gone through therapy too. And with the consent of their docs were able to actually stop the meds (they went into remission), after learning better coping and life skills in therapy. Their lives are relatively normal and very happy. Others I know, have to stick with their meds to maintain their ability to function (they have not experienced remission). I have taken meds in the past and had very bad experiences after sticking with them for a long time. I have also been to alot of therapy. I no longer take meds, and prefer to handle my symptoms with the tools i learned in therapy. Is my life normal? No, but it is mine and i am happy that i have chosen it. My boyfriend has never taken meds or had therapy, but is very well educated about his illness. He is very knowledge oriented and uses it to help him cope and control his moods. He was in remission for nearly a year, and now his symptoms are back on a much lower level. Is his life 'normal'? No, but he is happy with his choice and his ability to cope with his illness without meds.
Truth be told, part of 'normal'is not having a mental illness.Even remission is not an absence of the illness. So I think no, its not possible in the literal sense. What is possible though, is a relatively normal life and a happy/content one. Its up to you to decide how you want to achieve that and what it means to you.
I wish you luck.