I've been thinking about it for a few days and I still do not know. Which brings me to my answer: "Yes". and "No". Both yes and no.
Allow me to explain; Being that a man and a woman are both complicated to begin with not to mention very complicated to each other, how would we know if it was infact 'platonic'? What I think is that there are a guy and a girl who have much in common and let's say the girl completely honestly does not have physical desires toward the man. She never once had the desire to touch or hug him, or kiss him or whatever else desires could lead to. Now she is staying the path of a platonic relationship. But the man on the other hand has to use a snorkel in his dreams and thoughts because they are swimming with desire for this girl. There goes Plato's theory. But now you have to take into consideration something else. How would anyone know if the relationship fit the criteria? Well, you might ask each other. The only problem with that is not that one might LIE. BUT that one might not tell the truth. The reason being is that in that same relationship mentioned above, the girl approaches the man oneday and something like the following takes place between them;
Sally: Joe, I've been thinking about some of the things you say and do, and I wanted to make sure we understand each other in terms of our relationship with each other.
Joe: What sort of things are you talking about?
Sally: I don't want to make a big deal about it, but just the way you talk to me and remarks that make. And sometime how when you hold a door for me you will put your hand on my back when I walk by. I'm sure it's nothing but I wanted to make sure.
Joe: Oh, that sort of thing. Nothing to worry about Sally. I'm just trying to be a gentleman. It's surely nothing to worry about.
Sally: Ok, good, I didn't want our friendship to end because one of us started having different feelings about the other. I mean you and I are great friends, don't get me wrong, but we agreed way back when that we were friends and nothing more. Because we weren't interested like that. And you know I love you to death, like a brother!
Joe: I know Sally, nothing's changed. I wouldn't want to mess up our friendship we've been through too much and need each other's support..
Blah blah blah blah blah.. etc. and so on!
Now Joe didn't necessarily lie, but he surely wasn't truthful with Sally. What happened there was Joe indeed did have more than friends feelings for Sally, he was falling in love with her. But he knew that if he were to tell her that, that she would either turn and run or it would seriously screw things up for them. So he chose the easy path. He agreed with her to avoid losing her altogether. Smart move on his part for not losing her but he just lied, uh er, untruthed her. (Remember what the Bible says about a woman's wrath, Joe)
So do I think there could be such a thing as a platonic relationship? I'm sure there are many of them out there. But I would think they would have to be few and far between. One of the definitions of 'platonic' is: Free from physical desire. So even tho there are alot of people that we know who are friends and both are 'not married', there is something that does not qualify them as platonic you all parties were to be honest. Some say they don't 'fall for a guy' until they know his intentions. In other words if the guy is just a pal and being friendly because of some common interest or whatever, the girl maybe would hold her feelings back until she is sure how this guy feels about her. Which would be 'guarding her heart'... BUT she would admit that somewhere inside her are feelings of a non-platonic nature. (out the window again Plato!)
So to answer the question: In the grand scheme of things I would have to say, It is possible BUT not usually the case, whether admitted or not. Now that does not mean that they want marriage or anything.. but just that one of the two usually has desires hidden away and would admit if they were honest.
So in conclusion, it can be, but rarely. So guys buy an extra snorkel if you want, but if she asks, BE HONEST. And same to you girls, BE HONEST. Sometime being honest can be the most painful thing for both parties but it gives you a much better feeling when you close your eyes at night thant that bad feeling in your stomach when you know you lied to someone who loves you!