In an 'agile' process people should only be working on any particular 'story' for 3-5 day period. Everyone on the team should be interested in what they other people are doing, because they may in turn be pulled into work on some part of it next. In a healthy team everyone should be invested in the success of the product, and have some knowledge of most aspects.
Frankly, once things scale beyond that, where you have large teams of people disinterested in what's happening beyond their choice of specialization -- "agile" starts to not be a good match anymore.
