Can an infinite bandwidth signal be an infinite-time signal?
Let's consider the continuous time case.

Take a Gaussian $f(t) = \frac{1}{\sqrt{2 \pi}} e^{-t^2/2}$, which is positive for all time. 

The Fourier transform of a Gaussian is another Gaussian: $F(\omega)= c e^{- d \omega^2 }$ for some constants $c,d$  that you can work out on your own, so a Gaussian occupies all time and all frequency. That is, you can be both infinite in time and infinite in frequency. 

Your trade-off between resolution in time and frequency is given by the uncertainty principle. The Gaussian attains equality in the uncertainty principle. 

It seems like you're somewhat confused as to what Nyquist's sampling theorem says -- see this page for details (or your favorite signal processing text). You're assuming the signal is band-limited (i.e. is non-zero in some finite interval of frequencies) in order to be able to sample the signal at some prescribed rate in order to be able to perfectly reconstruct the signal.