Break a signal into segments, add them together and then perform an FFT on the result
That's not true (unless the signal is periodic, and those pieces are the base periods of it).

If you divide a signal $x_L[n]$ of a length $L$ into smaller pieces $x_i[n]$ of length $N$ each,

$$ x_L[n] = x_1[n] + x_2[n-N] + x_3[n-2N] + x_4[n-3N]$$ 

and sum them into a shorter signal $x_N[n]$ of length N 

$$ x_N[n] = x_1[n] + x_2[n] + x_3[n] + x_4[n]$$

and take $N$-point DFT/FFT of their resultant $x_N[n]$, you won't be able to reconstruct the original long singnal $x_L[n]$ back.

Indeed it's not even about FFT. You would not be able to retrieve the components $x_i[n]$ from their sum $x_N[n]$. So since there is no way to reconstruct back $x_L[n]$ in this manner.