There is a necessary and sufficient condition for a function $f$ of the real variable
to be "band-limited". It is called the Wiener-Paley theorem. $f$ must be a restriction on the real
line of an entire function of exponential type. Of course, here one implicitly assumes that
$f$ belongs to an appropriate space which permits to interpret it as a "signal".
For example, $L^2$ (finite energy), or $L^\infty$ (bounded amplitude), or Schwarz temperate
distribution etc.

P.S. This is a mathematical website, and the engineering terminology of the problem may
sound strange to some mathematicians. So let me try to translate:
We are taking about a function of a real variable for which Fourier transform is defined
in some sense. Band limited means that Fourier transform has bounded support, and
"time-limited" means that the function itself has bounded support.
Of course the function cannot be simultaneously time- and band- limited, unless it is zero.
This is a (very crude) form of the "indeterminacy principle", and also follows
from the Wiener-Paley theorem.