This appears to be such a common problem, that finding answer to my specific case is near impossible, because of the amount of noise in google or forum searches.
I have imported object to unity from blender. This objects had smaller objects inside it. Those smaller objects have animation on them, which was created in unity. If I open the animation window (Window->Animation) and press play - I can see correct movement of an object. Animation is set to play automatically in inspector, culling type is always animate. If I play the game however, animation does not play. Why?