This paper, http://www.academia.edu/8412078/Is_The_Nyquist_Rate_Enough, for one, refutes this.
More reading material:
EDIT: Your truncated sinusoids have to be harmonics of the total signal length for this to work trivially.
The first[1], the more theoretical of the two, is focused on digital communications where a Nyquist rate derived from the signal rate isn't enough to recover the signal function. The numerical examples given highlight recovering a sine wave from two, three, and six samples. The millions of samples in a single song might as well be infinite compared to these.
The second paper is much more practical and gives a number of real-world examples where naively applying Nyquist causes problems. The first, cutoff filters at half Nyquist to avoid aliasing, plauged many early CD players and other digital audio equipment. The second example is great as well, showing how understanding the signal being measured is important. The third example doesn't really apply to the subject at hand but is a good example of using aliasing for good. The fourth and fifth address the effect filters have on the signal. The sixth example refers to time response in control systems and minimizing phase delay to avoid oscillations.
[1]http://www.academia.edu/8412078/Is_The_Nyquist_Rate_Enough [2]http://www.wescottdesign.com/articles/Sampling/sampling.pdf