Absolutely it's possible to have a 'good eye' without knowing the technical skills. The same can be said of nearly any artistic pursuit, and many skill pursuits. In other fields, they call it 'a natural'. I've known people who have never painted in their lives, yet as soon as they are taught the bare minimum, they can turn out interesting, expressive, or brilliant pieces of art. Some folks who never took a class on drawing yet can do so with astounding accuracy. There are people who never sat behind the wheel of a race car in their lives, yet the first time they did so, were rivaling the speeds and controls of trained professional drivers. Many a shooting range has witnessed a first-timer pick up a gun and start smacking bullseyes with the greatest of ease.
With photography, it being an 'art' as much as or more than a 'science', there are those with natural artistic abilities or vision that can just instinctually know what will make a good photograph - feeling or judging the light, the color temperatures, the shadowplay, the geometrics and angles, how best to compose the photo in an interesting way - yet they know nothing of the technical side - how to set an ISO/Shutter/Aperture/White balance, etc. Even just seeing something that is ordinary to 99% of the people passing it by, and somehow noticing the extraordinary about it - maybe the angle, maybe the story, maybe the juxtaposition. I've seen brilliant photos of a piece of trash in a gutter - how many would have even known that could be a brilliant photo? All the technical knowlege in the world wouldn't have helped you judge that scene as something imminently photographable.
And there are very non-technical ways of still controlling a camera. I was a photographer since 1977 - but I first learned...really LEARNED...what the relationship was between ISO, shutter, and aperture, in 2003. I had never even heard of 'white balance' until 1997. And my primary style of shooting from 1977 to 1997 was sticking my Pentax ME Super SLR on 1/125 mode, and turning the aperture dial until the little green dot of the light meter hit the center in the viewfinder...and always using 400 speed film. I knew not why on any of those things. When I moved to digital in 1997, it was a revelation because I could see on the LCD screen what my shot would look like before I took it, and actually see the effects of various settings on the end result. But I still had no idea what shutter and aperture relationships were, despite completely manipulating and controlling my exposures...I simply used spot meter and P mode, and judged the metering of the scene by moving the spot meter around and locking it in when it looked right to my eye.
So yes...short sum up of the above - one can definitely have an 'eye' for photography, regardless of whether or not they have the skills and technical knowledge, and one can definitely possess high technical competence and still not have 'the eye'.