Band limited signal: Infinite time duration, time limited signal : infinite bandwidth
Here is an argument that doesn't require the uncertainty principle.

Assume $y(t)$ is a signal that is not identical to zero, with Fourier transform $Y(f)$. Let $s(t) = \text{rect}(t/T)$ a rectangular pulse of duration $T$ and Fourier transform $S(f) = |T| \text{sinc}(T f)$.

Then, $x(t) = y(t)s(t)$ is a time-limited signal with Fourier transform $X(f) = Y(f) \ast S(f)$. Since $S(f)$ has infinite support, so does $X(f)$.

Since any time-limited signal $x(t)$ can be defined in this way for a suitable choice of $y(t)$ and $s(t)$ (and a possible time-shift that doesn't affect the result), then we can say that all such signals have infinite bandwidth.

The same argument can be applied for the inverse Fourier transform: band-limiting a signal involves multiplication by a rectangular pulse in the frequency domain, or a convolution with a sinc in the time domain, resulting in an infinite-duration signal.