WASHINGTON—At Davis Elementary in this city's mostly poor southeast section, 178 students are spread out in a 69-year-old building meant to hold 450.

Three miles away, the new, $30 million KIPP charter school teems with 1,050 children. Toddlers crawl over a state-of-the-art jungle gym and older students fill brightly decorated classrooms. A waiting list holds 2,000 names.

Many...