If you look at the Wikipedia entry on the Nyquist Sampling Theorem, you should note that the summations to reconstruct the original signal go from negative infinity to positive infinity. In other words, that sum requires an infinite number of samples.

There are many signals of practical interest that can be approximately reconstructed with a finite truncation of the series. Note, however, that any signal that has only a finite length, eg has a uniformly zero amplitude after some time t_final, does not have a finite bandwidth, and cannot be exactly reconstructed by any sampling scheme. This is the case whenever you stop sampling a signal, eg it is always the case whenever you step outside the mathematical abstraction and start running real code on a real computer. So any signal reconstructed from samples is always approximate, except for some relatively trivial special cases.