There's a difference between "This survey is anonymous"/"Hey, Bob ranked his satisfaction with his manager as 1/5", and "This survey is anonymous"/"Here's a list of complaints from your team"/"Oh, this one is a complaint about how X impacts Y - I know Bob's the only one who's expressed interest in that in the past, I bet it's from him".
If my employer told me a survey was anonymous, I would expect anonymity on the Rank/Rate/True-False type questions, but assume that any freeform comments would be identifiable to someone as being from me based on style, content or something else.