At the risk of being redundant, I add yet another answer because I think that not all things have been addressed as clearly as necessary.
As mentioned in Dilip Sarwate's answer, the cross-correlation function of two signals $x(t)$ and $y(t)$ cannot vanish (for all values of its argument) if their spectra overlap, because this would mean that their cross-spectral density given by $X(\omega)Y^*(\omega)$ must equal zero. And the latter can't be true if $X(\omega)$ and $Y(\omega)$ overlap.
The other question is if the cross-correlation can be zero for zero shifts if the spectra overlap. This would mean that
This condition is called orthogonality. It can be satisfied for signals that overlap in time and that have overlapping spectra, if they - or their spectra - form a Hilbert transform pair. E.g., if $x(t)$ and $y(t)$ are a Hilbert transform pair then their Fourier transforms are related by
(This is what geometrikal addressed in his answer.)
The integral in (2) then becomes
which, if it exists (if necessary as a Cauchy principal value), is indeed zero, even though both signals overlap in time as well as in frequency.