If adamantium is not covering Logan's joints, can I cut his arms or legs there? Or even the neck?
According to this answer, his ligaments are in fact a weak point. And while it doesn't show him getting cut apart...
...Tearing him apart is entirely possible.
Note that this image is of Ultimate Wolverine, and not the main (Earth 616) Wolverine, who may or may not have sturdier ligaments.