A cold is a virus, so it really has nothing to do with binding.
That being said, it's sometimes hard to differentiate an actual cold vs. an upper respiratory infection, which is bacterial. Even so, binding isn't going to "cause" the cold to be located in your chest.
I'd say coincidence. The only thing I know of that is aggravated or possibly caused by binding is costochondritis, which is not cold-like. Either way, if you're really concerned, talk to your doctor.