Yeah, on spin-up the hard drive motors can pull up to 1A@12V each for a fraction of the second. This makes a big transient spike in power consumption. Big spikes in power like that cause the power supply's voltage to drop sharply, then recover. Certain power supply topologies, like double forward, have very good transient response and recover very quickly. Other technologies, like LLC resonant half bridge have poor transient response. Unfortunately all of today's highest efficiency power supplies use LLC resonant topologies (with few exceptions). This is good for efficiency, not good for transient response.
Now, poor transient response isn't a *huge* deal, but it can have an effect if you're trying to overclock. The transient load at startup can affect the voltage to your CPU, and could cause it to crash unless you give more voltage, heating up the chip and reducing its lifespan.
So you minimize the transient load by using "staggered spin-up", an option found on many RAID cards that allows you to start the hard drives one at a time so that instead of have 16A thrown at it instantaneously it sees several smaller 1-2A spikes that don't have nearly as much effect. This way you don't need to worry about the transient response of your power supply as much and can simply buy one with good efficiency and other electrical performance metrics such as voltage regulation and ripple suppression.