When a user searches for issues in a specific project (or a list or projects) and he does not have the Browse Project permission for the project, he gets the following message

The value '10001' does not exist for the field 'project'
Where '10001' is the project ID or name used in the JQL.

This message could be different for projects that do exist but the user simply does not have permission to browse it, like the following.
You do not have permission to browse the project '10001'

This would help a lot of customers that get this message and may believe the project has been deleted somehow.