Who is the company doing the trade-in? What country are you in? In the US, I always try to do any trade-in directly with Apple at an Apple Store, so they do the device inspection on the spot and you get either a gift card / trade-in credit on new purchase asap.
If you have to mail it in (even through Apple), they use a 3rd party company (depends on what country you're in), but 100s of posts about sketchiness with 3rd parties - horror stories of bait and switch (whether willfully or just incompetence and they don't internally track properly enough that your device was considered someone else's)...
Anyways, assuming it's a 3rd party trade-in company, I think much more likely that they are screwing up the inspection vs a "like-new replacement" (eg. official Apple refurb) you got via AppleCare.