Yes, you could, be it's not that ease.
You could set the boot order to boot from either hard drive, but that may not work.
During BIOS post, most motherboards allow you to choice which drive you'd like to boot to, by pressing a button during post. Mine is <F8>.
But the main problem would be the drivers from the hardware on the old drive with the old OS being put in the new rig, with different hardware. It could really freak out.
And then there is a licensing issue. Too many changes to a system require you to buy a new license (in simple terms, there are rules/execptions).