Yes, hangovers happen. Usually from drinking a lot of kava, sometimes from eating too much after drinking a lot of kava, and sometimes with bad kava (not noble.)
Even with a moderate kava session, I can sleep so well that the next morning I just don't feel like getting out of bed. Even if I don't feel awful---and in fact feel really warm and nice---it hangs over to the next day making it hard to get up on time. I got to the point where if I drank kava every night, every morning would be hard. Now I drink kava less frequently and don't have those problems as much.
One time I drank a bunch of kava with friends, went out and ate a huge barbeque dinner and felt like crap all the next day. I don't think it was the kava alone, but the combination of too much kava and too much food (I was stuffed at the end of the meal.)
Alternately, some kava (like Mahakea in particular) leave me feeling refreshed and happy in the morning and looking forward to my day.
There are a lot of variables in play, but yes, hangovers can happen. You'll have to adjust your consumption (timing, patterns, amounts, etc.) to find the sweet spot. There is DEFINITELY a sweet spot for me. Too little and I don't get much out of it, too much and I feel nauseated, shaky, crappy in the morning. YMMV.