The sights on my old j-frame revolver don't have quite enough lateral movement to allow me to center the groups.
I am thinking of taking the sight off and taking it apart so that I can:
1 - see if maybe it's just dirty or something, or
2 - see if maybe I can modify the sight blade or some other part to allow it to travel more.
Will these sights come apart? I've never tried before. Is this worth doing? Any other suggestions?

Obviousily we are discussing the factory adjustable rear sight RIGHT??? If so the little screw towards the front sight is all you need to remove and then slide the sight assemly to the rear to clear the frame. Lube and clean all. DO NOT disassemble the leaf portion as the screw is staked into the retaining nut. It is possible that the barrel as turned slightly in the frame and the barrel may need re-indexing by the factory. DO NOT TRY THIS AT HOME

Disassembly of the blade/adjustment screw requires the breaking of the screw. It is made with a fragile section that breaks when you attempt to unscrew it and hold the opposite, staked in side. Brownells sells the kit to change the sight blades if necessary.

Your barrel needs to be rotated the opposite direction that you have moved your sight over.

Bobby is correct. The sight windage screw has to be broken to disassemble the rear sight.

Thanks for the replys. Yes it is the rear factory sight. I don't remember using the gun for a pry bar, or dropping it. I don't see how the barrel could have turned.
Any way to check to see if it has turned? I only need to move the group about 2" to the right @ 25 yards.
I thought about filing the right edge of rear sight groove. This would widen the groove while moving the centerline of the groove to the right.

It sounds as if your barrel has loosened itself a wee bit, causing you to shoot left. If that is the case, you probably can see the front sight leaning to the left, as viewed from the front end of the UNLOADED GUN.

Perhaps try holding a straightedge vertically across the frame with the cylinder open, and view from the front end, using the straightedge as a reference to check the front sight for straitness. It sounds like your barrel needs to be turned in a wee bit more.

This is a tricky job that requires the right tools. It would be best to find a gunsmith that has a frame wrench to fit a J-Frame, or send it to Smith @ Wesson. Joe