I'm still under the impression that modifying and completely removing are different things, but perhaps it was a sloppy configuration or that they put too much trust in their employees.
Depends on the filesystem. Traditional unix permissions are simply r, w, x.
If you lack write permission on the directory, you can't delete any files in it, but you can still modify them if you have write permission on the files themselves. It's an unusual configuration, but it's possible with standard Unix permissions.
In traditional Unix you can delete a file if you have write access to the containing directory. The permission bits of the file don't matter.
On all (?) modern Unix variants, you can set the sticky bit on a directory to also require write access to the file.