I would venture the easy access to information typically increases the breadth of our knowledge more than the depth of any subject. In daily use of Google and the internet one comes across many facts normally outside your area of expertise. Typically one won't pause to study the specific topic but will remember about it. This allows you to go back and Google it when needed. Another example, you might Google how to set up something on Ubuntu but will probably forget the details once done. If the action needs to be repeated, you'll remember where to Google for it. I.e. your general knowledged increased, not so much your specialist knowledge.
Thus we become more 'generalist' but with a vast store of detailed info at hand.
If you want to focus on one specific subject, Googling will increase the depth of your knowledge, but this is a conscious act on your part. Akin to going to the library to study.
So Google is not making us more stupid. Rather it typically moves knowledge into a wider (but less detailed) domain.
This is actually a very interesting topic and we were discussing it in the office the other day. With Google and the internet one tends to want information fast and to the point.
I have personally found when reading the news that I merely scan articles for facts, names, dates etc. I no longer really read the whole article. I sometimes google a fact and get my answer from the search result page summaries without even clicking the link. When I read a story and it does not get to the point/facts within the first 2 paragraphs I get irritated and move to another story (unless it has a special interest for me).
Maybe it is a natural way of coping with information overload as we are bombarded with info from all sides.