If you set a form's "Anonymous form" property, then the form does not store the names of users. In a Notes client, a user can see in the document properties that no name is stored.
A suspicious user might realize that names can be stored in hidden documents, or be recovered from access logs. For example, this forum records the IP address that you used to ask your question, and this would still work with Anonymous web access.
So if there has been a complete breakdown of trust, users cannot trust your organization's IT systems.
I feel that a better way is to encourage users to use their names, and then demonstrate that nothing bad happens to those who risk asking awkward questions. Indeed, anyone who asks awkward questions should be positively and publicly rewarded.