OK, so let's go through a few troubleshooting things here.
Is this reproducible? Meaning, does this consistently happen when using the package feature? If you are using InDesign/InCopy in a standard server environment, does it still happen?
It sounds like everyone here is using some type of shared file storage such as Creative Cloud Files, or DropBox. Have you tried the same thing via e-mail to rule out the cloud storage? I tried to reproduce the problem on my end using DropBox but didn't encounter any issues.
Can you try to package a file with all of the files residing on your local hard drive and then package a file and sneaker-net it to another computer using a jump drive?
I know this may sound silly, but hopefully doing this testing can provide some insight into what is going on here. Let me know how you make out!
