There are various things to check:
1) Are they a member of a group with the relevant download permissions?
Admin > Groups > Folder Permissions
2) Perhaps they are trying to download a copy of the original, but they do not have permission.
Make sure the 'Can download original files' checkbox is ticked on the Folder Permissions page.
3) In Asset Bank you can dictate what is used to construct the name of the downloaded file. E.g. 'Original Filename', or 'Title'. If this has been set to something lengthy like 'Description' then there is a chance that the downloaded filename is too long for the underlying Operating System to handle.
Review this here:
Admin > Attributes > Display Attributes > Downloading