Break a signal into segments, add them together and then perform an FFT on the result If you break a 4 second signal into 0.5 second segments, add them together and then perform an FFT on the resultant 0.5 second data. 

Will the FFT / frequency spectrum be the simlar to an FFT on the original 4 second data?

This would mean that you can store long amounts of data in smaller memory if all you cared about was retaining the frequency components... I'm not sure if I am missing something though?
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.