How tightly is the left cup supposed to fit around a Shimano bottom bracket?
 

Though it wasn't broke, in violation of the old adage, I decided to replace a budget Chinese BB (part of several parts that were being transferred to a better frame). The new BB is a Shimano UN55 68x110, listed as NOS (new, old stock) from a reputable bike parts dealer up in N.C. (via ebay). It came in the original box, w/ paperwork.
Anyway, I noticed there was some play in how left cup fit around the BB, which struck me as odd. I checked the old Chinese unit I pulled, and the left cup on it fit pretty flush. But this Shimano unit there's just enough play that I can click it back and forth. It's been a while since I've had a Shimano BB in my hand, so I can't recall if this looseness is typical.
I installed it, and it seems to be a ok (whatever "ok" means in this case), but it's kinda nagging at me, so I thought I'd ask The Matrix (that means you guys ! :D)

How much play in mm?

Did you torque first the main BB body and then the NDS adapter to 50-70N-m as shown on page 17 of this document? https://si.shimano.com/pdfs/dm/DM-MDFC001-01-ENG.pdf

I thread the body finger-tight, then the adapter, then torque them in turn.

As I mentioned in a recent thread I do pretty much the same as dsb. So if the BB unit is tightened securely in the frame where's the slop coming from? It could be from the unit's bearing/adjustment (not that the UN55s have a bearing adjustment...). Typically many cartridge BBs tighten their bearing preload with the torqueing into the frame. Andy

No way to tell in mm's how much play there is. It's not a lot, but definitely enough that I could tell, and it struck me as odd. (and again, on the cheaper BB, the adapter slid on easily, but had no wiggle room.)
Also, I can't discern any movement now that it's installed (but you wouldn't really expect that, after the main body has been threaded in tightly. There could conceivably be some flex, but I wouldn't be able to detect any back-and-forth rattle.).
And, yes, I did install the main (right side) piece first, followed by the left-side sleeve (aka 'adapter"). But, I don;t own a torque wrench, say I've always done these by feel and sight.

Confusion here. The play is able to be noticed after all is tightened properly?? Or was the play noticed before complete tightening? Andy

Or when you had the pieces in hand before installation?

Yes, I'm talking about in-hand, prior to installation. Once the right side (main body) is tightly threaded into the BB shell, there's not likely to be enough detectable movement to tell, even if it's there. My concern is that the BB may not be fully supported on the left side, (essentially free-floating inside the left cup), resulting in a constant flex with every pedal stroke. Last night, I pulled the whole unit again, and there's definitely a noticeable gap between the BB body, and the left cup, that seems abnormal going by memory of others, and that isn't replicated in the cheaper BB (which I also checked again, it has a pretty solid fit).
So, I don;t know if this slightly loose fit is typical or not, and if it's a concern. If it is abnormal, there's also the question of how it happened. It's not damaged or mis-shaped, it threads into the BB shell perfectly, and I don;t think it's possible it's the wrong piece from another sized BB (so I keep hoping one of the other forum gurus will chime in that this is perfectly normal).
BTW, back when I first noticed this, I also tried the left cup from the cheaper unit (thinking I might just keep that piece), but it also fit loosely on the Shimano body, so (assuming there's an abnormality here), the issue isn't actually that the left cup is large, but that the Shimano BB body is a smaller diameter.

The first time I replaced a cartridge BB, I noticed the same play. I got some creaking when pedaling after installation. I was living in a remote town before the days of Amazon Prime, so I just shimmed it with a piece of metal foil chimney tape. It worked perfectly for the next decade or so. I've done dozens since then and have never again seen the same issue.