It's also possible to develop patches on the back of the throat from a virus. Looking in my kids' mouths was the first thing I did when they got a sore throat. If I saw patches, I took them in. A couple times it turned out to be a virus that ran its course. Huh, what a nasty trick.
Don't forget some white patches can be thrush.
It is possible. One of my children never got a fever with strep. Would just feel tired and throat hurt. When you looked at throat you could see white blotches and then we knew for sure.