Nyquist Frquency isn't what it is supposed to be
The closer the sampling frequency gets to half the (highest spectral) frequency of the signal, the longer one has to sample to get a sufficient bounded fit (for some bound) of a reconstruction using those finite number of samples.  This required sampling time extends to infinity as one approaches Fs/2.  

So, you simply didn't sample for long enough in your experiment.  Try 5 thousand and 5 million samples for sampling a 12.9 Hz sinusoid at 26 sps, and you will see a closer fitting looking plot over most of an X axis of 5 thousand or better million reconstructed sample points.  Then try a sinusoid even closer to Fs/2 and you will find you need to sample even longer to get the same visual quality of fit.