More specifically than the permissions GPGeorge suggested, it is mandatory that all users of an Access database have the Windows high-level permission MODIFY on the database file itself and the folder in which it is located. High-level MODIFY covers some lower-level permissions (which you can see by going "advanced" on the permissions dialog). Having the high-level MODIFY setting is sufficient and necessary for proper operation of the database. The next higher high-level setting is FULL CONTROL, but that is NOT necessary for Access.
There are also multiple sources of permissions. If this is a formal domain, there will be domain-level permissions. If not, there will still be network-level permissions. There is also the matter of local registry settings, which includes but is not limited to the Trusted Locations list. To properly investigate the user's problem, you need to do an experiment with that user and with a user who is NOT having problems. This is the experiment:
1. From a user's computer, find the BE file that they all share using the Windows File Explorer window. Right-click on the BE file to get Properties, then select the Security properties from that complex dialog box.
2. From the security dialog box, check Advanced security. There is where you would see the high-level or advanced security settings. Take notes.
3. There is a button for "Effective Security" which tells you that user's permissions AFTER taking into account all of the layers of security. Take more notes.
4. Now repeat these steps for the folder in which the BE file is stored.
5. When done, you will have BE permissions and effective BE permissions for two users - one who CAN use the DB and one who can't. Compare them for any differences.
6. The symptom you describe USUALLY means you cannot create the .LDB or .ACCLDB file, which is usually a folder problem. (Not 100%, though.)
7. The question that also needs to be answered is whether the failing user can SOMETIMES use the DB or NEVER can use the DB. The "cannot lock" message might ALSO be due to the .LDB or .ACCLDB file having incorrect permissions, thus blocking a user from updating an existing lock file.
To eliminate the possibility of this being caused by FE file corruption, can you provide a copy of the FE from a user who CAN use the DB and transmit that copy to the user having difficulties?
