Is it possible to reduce a 1" thick board to 1"/N boards?
Theoretically, yes. Reasonably? That depends. You're describing what's termed "resawing", for obvious reasons. And you're absolutely correct on the basics: a properly tuned bandsaw with enough room for the width of the board you're resawing, the right blade, and typically the saw's fence (or a homemade jig) to keep the board properly on it's edge, at 90 degrees to the table.

So in practice, especially when you first start learning this skill, what happens a lot is that the blade will start cutting off to one side or the other. So you're constantly trying to steer the blade and the workpiece to bring the cut back to center. You end up turning one 1" board into 2 boards that range from 1/4" in some spots to 3/4" in other spots. Then you run these wavy-surfaced boards through the thickness planer a few times, and you end up with two 1/4" boards and a whole lot of sawdust.

If you're trying to build your skills, then I don't mean to discourage you. It's an important skill to have if you're gonna do some advanced projects (Google "bookmatching" for example). But if this is a one-off project for you, I'd recommend repurposing the already purchased wood and buying the actual thicknesses you need from a local hardwood dealer (i.e. not HD or Lowe's). Or as another response above mentioned, you could at least thickness plane this wood down and use it for your 1/2" needs. Good luck!