These pictures show the gap between my counter and the wall, and re-caulking is only part of what I want to do. It was suggested to me that the cabinet is coming apart from the wall and I should try to see if I can somehow tighten the bolts that hold it to the wall, but I can't really see where those are. If I try to pick up or push the cabinet it doesn't really budge. Any ideas?
The screws attaching the cabinet to the wall are placed at every stud, near the top of the cabinet where you see the solid wood. Studs are typically spaced every 16" and can be found with a stud finder. The cabinets will be attached to each other, so look for screws in adjacent cabinets, too, tightening everything uniformly. There should also be something holding the cabinet down to the floor, behind the toe kick on the front.
What you need to find out is why the cabinet is pulling away. Perhaps it's just caulk drying and shrinking. It's possible your floor is settling/sagging in the middle of the room. Or the outside walls could be going out of plumb. Some of these, like sagging floor joists, could indicate other problems that need to be solved first.
To fix the problem, you could remount the counter top tighter to the wall. But 99% of us would simply add more caulk.