How to use responsive design with bootstrap views
I think your problem is in using Views grid format. That is table based and not responsive in the way you would want it. Tables can be responsive but in a table way (hiding of columns and inserting sliders etc). 

You need to use 'unformatted list' as the format, and then for your 'row class' you need to set appropriate bootstrap col-* classes and you have to get a bootstrap row class in there as well. To do that your best bet is to override the views template files and preprocess functions.

There are plenty tutorials etc available for that on the web.

Bootstrap provides the framework, you just need to make sure your html conforms and that you put the classes you need in the right places. It can be done in views without any major issues.

You should also look at these modules as they definitely will help you (provided you have read the twitter bootstrap documentation and understand at least the basics)

Views Bootstrap, Display Suite Bootstrap Layouts, Panels Bootstrap Layouts