A server fighting a sleeve of cups one-handed and a delivery arriving with lids off are usually the same defect wearing two costumes. It is the rim.
What a rolled rim is for
The top edge of a moulded cup is curled back over on itself into a small bead. That bead does three separate jobs: it holds each cup a fraction clear of the one below it in a stack, it gives a lid skirt something to grip, and it stops a cut edge meeting a lip.
Rolling it takes tooling time and a moment of cycle. Skipping or shortening it saves a supplier money on every single cup.
Failure one: the solid column
Without a bead, cups seat into each other. A sleeve stops being a stack and becomes a single plastic column that has to be twisted apart with two hands.
In a queue, at a bar, that is seconds per serve times every serve of the night. Nobody logs it as a product fault; it just makes the bar slow.
Failure two: the lid that lets go
Lids do not clamp the cup wall, they grip the rim bead. Under-roll it and the lid has less to hold, so it seats without a proper click and lets go at the first knock in a delivery bag.
This is why lids must be matched to a rim diameter and roll profile rather than to a size name on a carton.
The two-second audit
Run a thumbnail round the top edge of a cup. Smooth bead: rolled. Distinct edge you can catch a nail on: not.
Then seat a lid and turn the cup upside down over a sink. Those two checks, on a sample, before the pallet — and both failures stop being your problem.