Clarify excerpt from Feynman Lectures on rotations in three dimensions In The Feynman Lectures, vol. I, chapter 20 Feynman discusses rotations in three dimensions and explains angular velocities may be added as vectors; in particular he says:

What about angular velocity? Is it a vector? We have already discussed
  turning a solid object about a fixed axis, but for a moment suppose
  that we are turning it simultaneously about two axes. It might be
  turning about an axis inside a box, while the box is turning about
  some other axis. The net result of such combined motions is that the
  object simply turns about some new axis!

Is this actually true? Take a look at the following gimbal animation from the Wikipedia; suppose the green ring is the object turning about an axis inside a box, represented by the external ring; does the green ring simply turn about some new axis?

 A: The example given is certainly misleading. Rotation in 3D is hard to visualize. In the case of the green ring, you can say it has an instantaneous axis of rotation which is the vector sum of the rotation about the two axes at a specific moment in time.
But the support structure of the green ring itself rotates about the vertical axis, and so the instantaneous axis about which the green ring rotates does change. As a result the sum vector will precess - the ring does not simply rotate about one fixed axis. But at any one moment in time the motion of each particle looks as though they all rotate about one axis (as they must: the center of mass is fixed and so is the relative distance between the points; the only motion allowed is rotation and that can only happen about one (net) axis at a time.
Incidentally that motion is very much like the motion of a coin spinning on a table: at any given moment it spins about an axis that goes through the contact point with the table and the center of the coin, but while spinning it experiences torque from gravity which causes the axis of rotation to precess. The net result is the wobbly motion you observe (it was a famously difficult problem in my undergrad mechanics class to analyze that motion and deduce from it the rate at which the head on the coin appears to spin... )
A: I guess he was talking about rotation through two stationary axes, which makes the box example incorrect. If the inside rotation axis rotates, there's no reason for it to be a constant rotation.
That being said, if you first rotate through one axis and then another, the combination will give you a rotation through a third axis. If you want the math behind it read about the 3 dimensional rotation group SO3 .
