The idea of dual springs in case one breaks was an older belief which had little practical application in engines that do more than a couple of thousand revs. In reality it is generally the main spring that breaks and the inner spring is too weak to control the valve which is going to bounce into the piston anyhow.
When a valve spring is operated it will start to oscillate/resonate/vibrate. At a given frequency the valve spring will lose control of the valve which will start to bounce off the seat and to spin. The frequency at which the valve spring oscillates varies depending on the diameter and winding and strength of the spring. In order to reduce the effects of this some valve springs are fitted with dampers, often a steel sleeve within the spring which rubs against the spring. Dampers can cause cause an excessive heat build up in the springs, but they do help damp out that oscillating effect. Another way is to use double or better still triple valve springs. Each of these springs will have a different frequency at which it will oscillate and as such one spring will help cancel out the effect of the other. The friction caused by the rubbing of the springs against each other in double and triple springs also helps to damp out this effect. A dual spring having a combined seat pressure of say 100lbs will always outperform a single spring having the same 100lbs seat pressure.
Modern spring development has resulted in beehive and conical springs. A conical spring tapers from bottom to top whereas a beehive spring only tapers in the top couple of turns. These springs damp out the oscillating effect because the different outside diameters of the coils have a different frequency at which they oscillate, so one coil within the spring helps to damp the effect of another.
There have been major advances in valve spring technology (double and triple springs as well as the conicals and beehives mentioned above) in the past few years which has resulted in Nascar and other "pushrod" V8 race car engines being able to increase safe rev limits from 8000rpm to 10500rpm. The old idea of ever stronger springs to stop bounce is not necessarily the right approach, but rather better designed springs in which the oscillation is minimised at an appropriate seat poundage rate and of course a lighter valve train. The camshaft design is ultimately also responsible.
The attached link shows a dual valve spring in slow motion. A single spring would oscillate more. See the oscillation effect up and down the spring, and a bit later in the video see how the valve is bouncing off the seat. A driver/rider would not be aware of this valve bounce but it would soon lead to failure.https://www.youtube.com/watch?v=_REQ1PUM0rY
The valve bounce that is noticed by the driver/rider is when the valve is lofted off the top of the cam lobe and is not controlled on the closing of the valve by the valve spring holding the follower against the cam lobe. In this case the spring eventually brings the valve train back in contact with the cam part way down the closing ramp, at which point it will probably bounce again, and then again when it hits the seat. Piston to valve contact is often the result.