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
$$\int_{-\infty}^{\infty}x(t)y^*(t)dt=0\tag{1}$$
Or, equivalently,
$$\int_{-\infty}^{\infty}X(\omega)Y^*(\omega)d\omega=0\tag{2}$$
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
$$Y(\omega)=-j\cdot\text{sign}(\omega)X(\omega)\tag{3}$$
(This is what geometrikal addressed in his answer.)
The integral in (2) then becomes
$$j\int_{-\infty}^{\infty}|X(\omega)|^2\text{sign}(\omega)d\omega\tag{4}$$
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.