I'm somewhat of a contrarian on this matter. I think the common factor among nearly all associates at the most prestigious firms is academic achievement. Most come from top schools, but those that don't did well at whatever school they attended. So, study hard and get good grades. I think the government typically cares about grades and extracurricular activities, in addition to other key factors in your resume (previous work experience, mostly). Big Law doesn't want to hire someone who is unlikely to pass the bar, so I think they tend to focus on the grads from the top schools because they can tout their resumes to their clients and because there is little risk that those grads won't pass the bar. For government, you will almost certainly need to be a licensed attorney with relevant experience and good grades. The school you come from is probably less relevant than what you have done while you were there. I'm sure plenty of posters will disagree with me on this. I am also positive that the importance of the school you attend will greatly differ from agency to agency.
As far as networking and local connections go, many jobs are filled by word of mouth and recommendations from associates and colleagues. Developing networking skills is a must for all professionals. Don't make this an either/or proposition. Learn sound networking skills AND go to a good school. That is your best bet. If you can't get into a great school, do well at the one you choose and network your a$$ off either way.