Possible to fry a circuit with a power supply under its current requirement?
This is just a hypothetical scenario.  Don't know if it applies to hard disks.

Suppose there is a control loop in a system, which actively keeps the system out of some harmful condition.  Due to insufficient supply current, the supply voltage sags (as mentioned in the O.P.).  Due to the sag in supply voltage, control system can't work properly.  It can't keep the system out of harmful condition any more.  So, the harmful condition occurs and does permanent damage.

I'd say that something like that would be a design flaw.

P.S.  Since O.P. is about PC components, I wonder if it's more of a serverfault question?