Few years ago, when I was walking to the university I had a distinct feeling that I was being watched. I stopped and looked around, but there was no one around. Then I noticed a bush next to me. Inside was a small owl and it was staring at me like owls usually do. We stared at each other for a few moments until I continued my journey.
I've heard it said that our eyes take in more information than we consciously process (same thing is true with hearing). You may have spotted the owl out of the corner of your eye, and only subconsciously registered it, strong enough that it eventually triggered your conscious mind to feel like you're being watched.
I saw a PBS nature program about crows. It was fascinating
(with all due respect to Mr. Spock). The crows can recognize people from very far away, and have a "communal" understanding of who is dangerous and who is not. If one spots someone potentially dangerous (resembles someone who previously endangered them), they'll rapidly communicate with each other about the danger then deliver vicious call sounds to the potentially hostile intruder.
Animals are generally more aware and knowledgeable than most people realize.
Yesterday I was walking around outside and I saw what appeared to be a black stray cat. It started to scuttle across the street but looked both ways first, and when it got midway it paused, waited for a car to pass, quickly looked both ways again and then darted across the rest of the way to safety. I've seen human beings far less observant of traffic when crossing the street!