Extrapolation after taking inverse Fourier transform
An FFT possesses linear features. Let us use a linear analogy. If one tells you that the sum of two numbers is $a+b=3$, what can you say about these numbers? Not much. More generally, $n$ linear equations on $m$ variables is underdetermined when $n&lt;m$: you do not have enough information to retrieve the variables without more conditions. This can be experimented with a very simple DFT case.

Let us assume the vector $x = [1,-1]^T$. A simple two-point DFT is defined by the matrix:

$$ \begin{bmatrix}
 +1 &amp; 1\\ 1 &amp; -1 
\end{bmatrix}$$

Its products with $x$ yields either $0$ (for the DC) or $2$ (up to a  $\sqrt{2}$ Fourier definition scaling) at Nyquist. Can you recover the two elements from $x$ from only a single coefficient $0$ or (xor) $2$?

Probably not, without further assumptions, or on very specific signals.
This happens whenever you have less DFT points than samples. There is no bijection between two finite sets of different sizes. This is indeed one definition of infinity: a set is infinite if it can be put in bijection with a proper subset. 

However, using sparsity, positivity properties, there are frameworks to retrieve (at least approximately) the data.