Perfectly isn't hyperbole here. That is mathematically shown.
And the part of perfect reconstruction that nearly everyone forgets is that it requires an infinitely long sampling of an infinite-time signal. If you use a time-limited sequence, you do not get perfect reconstruction. I've been in the business of signal sampling and reconstruction a long time, but I'm still having trouble finding an instance of a sampling that spans infinite time (for those who are humor impaired, that was a joke).
More practically speaking, because all digitizations that you come across, or design, have finite time span, it means that the reconstruction accuracy starts to get worse and worse the closer you get to the Nyquist threshold, and the effect is worse and worse the shorter the sequence length. Here's an extreme thought experiment: you have two samplings of a signal just a hair below Nyquist. Just a hair below. The sequences are both pretty short, say 4 samples long.
In the first sampling, you got unlucky, and the samples all happened very close to the zero crossings so that all of the quantized values are 0. Reconstructing that yields a DC value of 0V.
In the second sampling, you got lucky to the opposite extreme, and the samples all happened very close to the peaks, so that the quantized values are alternately +PEAKVALUE and -PEAKVALUE. Reconstructing that yields the original sinusoid at the intended amplitude.
Which sampling is correct? If I just give you the sampled values, there is no way to tell. Any reconstruction from amplitude 0 to amplitude PEAKVALUE would be accurate, and there is no way of knowing for sure what the phase was.
Now, if the sequences were infinite length, then, eventually, no matter how fine that hair was below Nyquist, you'd start to see the beating against the sampling clock, and, eventually, be able to observe samples that spanned the entire range of the sinusoid, making accurate reconstruction possible, including phase. But, again, you'd need an infinite sequence, with an infinite sinusoidal signal.
What are the real-world consequences of this problem? (1) you lose phase and amplitude information of the original -- they CANNOT be reconstructed accurately -- as you approach Nyquist, with the effects getting more and more pronounced as the sequence length gets shorter and shorter. (2) If you really want dead-on accurate reconstruction up to a frequency F, you should be sampling at 5F, not 2F. That also gives you more room to design good anti-aliasing filters on the sampling side, and carrier frequency filters on the reconstruction side.
Remember, people, Nyquist is the mathematical limit, not the practical, usable threshold.