Just setting the name style to first, middle, and last name does not mean they've a middle name stored in the database. If they don't supply a middle name they won't have one. Edit the user in CB > User Management and verify they've a middle name. The [name] substitution pulls its value directly from Joomlas _users table where you can also check to see if they've a middle name stored.

Does the middle name exist in the name column of _users in your database? The [DETAILS] usage uses [name], which comes directly from Joomla name column. The [firstname], [middlename], and [lastname] usages come from CB directly since Joomla doesn't split the name in storage.

Strange, I'm not sure what to suggest. As [name] comes directly from the Joomla stored name column in _users it should be there. I suppose you could try [firstname] [middlename] [lastname] to let the values come from CB instead. [name] seams to be working fine in my tests. Try substituting [name] into a Custom HTML field and see if it displays on their profile correctly.