There are 2 project records that a user is trying to access. Project 1 has read-only access for a particular security group (Security tab of the project). Project 2 security does not have read-only access (Security tab of the project). The user in the security group logs in and accesses Project 1. The record is read only, which is correct. Now, the user accesses Project 2, which should be editable. It is not. It is still read only.
Moving forward, an issue that could cause user permissions to be reduced to read-only after viewing a read-only project record has been resolved. A previous workaround of clearing the security cache to reset the permissions level is no longer necessary.