If an object is increasing speed at a constant rate around a circle, does the magnitude of acceleration still change since there are both the tangential acceleration component (whose magnitude is constant) and the centripetal component (which changes in magnitude as the object increases in speed)?

The tangential and the centripetal component are at right angles.
The tangential component is constant.
The centripetal component increases.
Obviously the magnituded of the total acceleration also increases

If the object is accelerating at any rate then the magnitude of the angular, tangental, and centripetal components of the objects motion will also change. All the components of the ojects motion will change because they are all describing the same motion of the same object even if we as humans split its motion into many component motions on paper so we can fully describe in mathmatical terms. My point is that when an object is in circular motion you can not change one component of that motion with out affecting the other components of its motion. Centripetal Acc. is based on Tangental speed and tangental speed is based on angular speed which is based on the measurement of angular displacement in a certain time interval which is simply motion in a circle. Take a close look at the angular kinematic equations and try to see how they all fir together. Please post the scenario thats giveing you troubles if this spiel doesnt help.