Yes,
White noise is kinda of like white light. White light is made up of every color of the spectrum. White noise is made up of every frequency in the electromagnetic spectrum.
You cannot totally eliminate noise, even by filtering(even if the filter is ideal), because you can't eliminate the noise that is at the same frequency as your audio. You can help prevent noise with well shielded cables and by using a power conditioner though.
So yes there will always be a little noise no matter what you do, it will be even more noticable when you crank the amp, because you are amplifying that base noise.
Single coil pickups are usually a little more noisey than humbucklers, but not always.