If you connect shields at both ends for a high speed(*) unbalanced digital signal cable, you will get
a lot of problems with induced noise spikes being injected into the grounds, which for an unbalanced
signal can have similar effects as if you put it on the signal wire. I am talking high frequencies here
where the stray inductance dominates the situation.
Induced voltages are the same for all conductors in the cable, but the current depends on the
impedance of the conductor going into the equipment - typically the shield is the lowest impedance
carries significant current - connecting this to ground at the input of a circuit will play
havoc with that ground voltage relative to other parts of the circuit at short time-scales if there
are noise spikes around.
You often see ferrite beads on a cable at each end, these help block induced currents on the shield
as well as reducing RFI emitted by the cable. Ferrite acts as a high impedance at high frequencies,
but only to the common-mode currents, differential is unaffected.
The other function of a shield, to protect against capacitive pickup, is completely different, and
only requires connection in at least one place to ground. It too can inject currents into ground.
Best practice is to ground the shield at the transmitting end, but not the receiving end (an RFC
can be used for DC continuity). Bidirectional cables are more problematic and the real solution
is to use differential signalling (like USB, Firewire, ethernet, LVDS video cables, etc etc).