The brief
The X1 had been bouncing in the rear after bumps, the back end carrying on moving instead of settling, with reduced stability through corners. Those are the signs of worn rear shock absorbers. A shock absorber controls how the spring moves so the body settles after a bump instead of bouncing, working by pushing oil through small valves inside it. When the seals wear and the valving tires, it loses that control, so the rear floats and bounces over bumps and feels loose in a corner because it isn't being kept planted. And the top mounts the shocks sit in wear with them, so on a set job they go in together. Worn shocks only get worse, and they take the ride and the stability down with them.
The diagnosis
On the lift the rear shocks were checked, and bouncing each rear corner by hand confirmed both had lost their rebound damping past spec, with the top mounts worn too. One side was no better than the other. When both are tired together you do them as a set, shocks and mounts, fitting one fresh against a worn one leaves the rear behaving unevenly, so it was the full rear set.
The work
Both rear shock absorbers came off, and a matched set of genuine BMW-spec shocks went on with new top mounts, every fastener torqued to the manual figures. A road test confirmed the bouncing was gone, the ride had settled, and the rear stayed planted through corners.
The outcome
No bounce over bumps, a settled ride, and the rear planted through corners. The X1 went home with the rear suspension back to how it should ride. Shocks and their mounts wear together, so a fresh matched set brought the rear back to spec rather than chasing it part by part.